19. Waves

YaleCourses

22 Sept 200871:54

EducationalLearning

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TLDRIn this educational lecture, Professor Ramamurti Shankar explores the fundamental aspects of wave physics, focusing on wave equations and their properties. He begins by discussing the derivation of the wave equation for a string under tension, illustrating how tension and mass per unit length relate to wave velocity. The professor then delves into solutions of the wave equation, explaining the significance of wave forms like A cos(kx - ωt) and the relationship between angular frequency (ω), wave number (k), wavelength, and time period. He further investigates wave energy, power, and the concept of intensity, particularly in the context of sound waves. The lecture also covers the Doppler effect and its implications for both sound and light, and concludes with an in-depth look at wave interference, including the physics behind the double-slit experiment and the behavior of waves in tubes of varying ends. This comprehensive overview provides a solid foundation for understanding wave phenomena in physics.

Takeaways

📚 Professor Shankar begins the lecture by discussing the properties of waves, specifically using the example of a string under tension to illustrate wave behavior.
🌌 The wave equation derived in the lecture is \( \frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} \), where \( v \) is the wave velocity, and relates the second spatial derivative with the second temporal derivative of the wave function \( \psi \).
🔍 The lecture emphasizes the physical interpretation of the wave equation, highlighting that it represents the force balance on a small segment of the string and is analogous to Newton's second law, \( F = ma \).
📉 The concept of wave solutions, such as \( A \cos(kx - \omega t) \), is introduced, and the conditions for these solutions are explored, including the relationship between angular frequency \( \omega \) and wave number \( k \).
🌀 The lecture explains that any function of the form \( x - vt \) is a solution to the wave equation, indicating the wave's propagation and the significance of \( v \) as the wave's velocity.
🔄 The properties of wave interference are discussed, including constructive and destructive interference, and how they relate to the amplitude of the wave at different points.
🎵 The relationship between wavelength (λ), frequency (f), and wave velocity (v) is detailed, with the formula \( v = \lambda f \) being a key takeaway.
👂 The Doppler effect is explained, showing how the perceived frequency of a wave changes for an observer depending on the relative motion between the source and the observer.
📊 The lecture touches on the calculation of energy in a vibrating string, defining energy per unit length and discussing the implications for the power required to maintain the wave.
👨‍🏫 Shankar uses the example of a piano tuner to illustrate the concept of beat frequencies, which occur when two waves with slightly different frequencies interfere and create a pulsating sound.
🎶 The lecture concludes with a discussion of wave interference in the context of musical instruments, specifically how standing waves form on a string fixed at both ends, leading to the production of harmonic frequencies.

Q & A

What is the significance of choosing a string under tension as the medium to study wave behavior?
-A string under tension provides a simple and concrete example to study wave behavior. It is an idealized model that allows for the mathematical analysis of wave properties such as displacement, frequency, and the wave equation without the complexity of a three-dimensional medium.
How does the wave equation derived from the tension and mass per unit length of the string relate to Newton's second law?
-The wave equation T(d^2ψ/dx^2) = μ(d^2ψ/dt^2) is derived from considering the forces acting on an element of the string. It is analogous to Newton's second law (F=ma), where T is the tension (force), μ is the mass per unit length, and the second derivatives represent acceleration.
What is the small angle approximation used in the context of the wave on a string?
-The small angle approximation is used to simplify the mathematical analysis. It assumes that for small angles, sin(θ) ≈ θ, cos(θ) ≈ 1, and tan(θ) ≈ θ. This approximation is valid when the displacement of the string is small compared to its length.
How does the curvature of the string relate to the net force acting on a segment of the string?
-The curvature of the string is related to the net force through the second derivative of the displacement function. A positive curvature indicates that the string is below its normal position and experiences an upward force, while a negative curvature means the string is above its normal position and experiences a downward force.
What is the physical interpretation of the solution A cos(kx - ωt) to the wave equation?
-The solution A cos(kx - ωt) represents a wave with amplitude A, wave number k, and angular frequency ω. It describes a wave that propagates in the x-direction with a velocity v = ω/k, where v is the wave speed.
What is the relationship between the wavelength λ and the wave number k?
-The wave number k is the inverse of the wavelength λ, multiplied by 2π. This relationship is expressed as kλ = 2π, which means that when the position x increases by one wavelength λ, the angle in the cosine function increases by 2π.
How does the time period T of a wave relate to the angular frequency ω?
-The time period T is the time it takes for the wave to complete one full cycle. It is related to the angular frequency ω by the equation ωT = 2π, which means that ω is the angular frequency in radians per second and T is the time period in seconds.
What is the significance of the wave velocity v in the context of a wave on a string?
-The wave velocity v is the speed at which the wave propagates along the string. It is determined by the properties of the medium (the string in this case) and is given by v = √(T/μ), where T is the tension and μ is the mass per unit length of the string.
How does the energy per unit length of a vibrating string relate to its amplitude and frequency?
-The energy per unit length (u) of a vibrating string is given by u = ½μA^2ω^2, where μ is the mass per unit length, A is the amplitude of the wave, and ω is the angular frequency. This shows that the energy is proportional to the square of the amplitude and the square of the frequency.
What is the concept of intensity in the context of sound waves and how is it calculated?
-Intensity is a measure of the power of a sound wave per unit area. It is calculated by dividing the power (P) by the area (A) through which the sound wave is passing. In the context of a point source emitting sound waves, the intensity at a distance r from the source is given by I = P/4πr^2.
How does the Doppler effect influence the observed frequency of a wave?
-The Doppler effect causes a change in the observed frequency of a wave when there is relative motion between the source of the wave and the observer. If the source is moving towards the observer, the observed frequency increases, and if the source is moving away, the observed frequency decreases. The change in frequency is given by f' = f((v - u)/(v + u)) where f is the source frequency, v is the speed of the wave, and u is the speed of the source relative to the observer.

Outlines

00:00

📚 Introduction to Wave Mechanics

Professor Shankar begins the lecture by discussing the concept of waves as disturbances in a medium, using a string under tension as an example. He introduces the wave equation, which relates the second derivative of the wave's displacement with respect to position and time to the square of the wave speed, v. The wave speed is determined by the tension in the string and the mass per unit length, represented as v^2 = F/μ. The professor emphasizes the physical interpretation of the wave equation as a form of Newton's second law, F = ma, where the force on a small segment of the string is equated to the mass of that segment times its acceleration. He also explains the small angle approximation used in the derivation, which simplifies the trigonometric relationships in the wave equation.

05:03

🌊 Deriving the Wave Equation and its Solutions

The professor revisits the derivation of the wave equation, emphasizing the cancellation of dx and the simplification to the standard form. He then explores the solutions to the wave equation, specifically the form A cos(kx - ωt), and explains how this form satisfies the wave equation under the condition ω = kv. This leads to a discussion about the physical meaning of the constants A, k, and ω, with A representing amplitude, k related to the wavelength (λ) through k = 2π/λ, and ω being the angular frequency related to the time period (T) of the wave. The lecture also touches on the limitations of the small displacement approximation and its implications for the amplitude of the wave.

10:04

🔍 Analyzing Wave Properties: Wavelength, Frequency, and Velocity

In this section, the professor delves deeper into the properties of waves, focusing on the relationship between wavelength (λ), frequency (f), and wave velocity (v). He explains that the wavelength is the distance between successive peaks or troughs of a wave, and it is related to the wave's frequency by the formula λ = v/f. The angular frequency (ω) is introduced as 2π times the regular frequency, and the time period (T) is defined as the reciprocal of the frequency. The professor also discusses the physical interpretation of these properties, such as how the velocity of a wave on a string is determined by the medium's properties and is independent of the wave's amplitude or frequency.

15:05

🚀 Energy and Power in Wave Propagation

The lecture shifts focus to the energy and power associated with wave propagation. Professor Shankar calculates the energy per unit length in a vibrating string, using the analogy of simple harmonic motion to find the kinetic energy when the velocity is at its maximum. He introduces the concept of energy density (u) and explains how the power supplied to maintain the wave is the energy per unit length multiplied by the wave velocity. This leads to a discussion of the unique property of one-dimensional waves, where the amplitude remains constant as the wave travels, unlike in three dimensions where the intensity of the wave decreases with distance from the source.

20:07

👂 Intensity, Decibels, and the Doppler Effect

The professor introduces the concept of intensity, which is the power per unit area, and explains how it is used to measure the loudness of sound in decibels (dB). He discusses the logarithmic scale of decibels and how it relates to the human ear's perception of sound intensity. The Doppler effect is then explored, describing how the frequency of a wave changes for an observer when the source is moving towards or away from them. The formula for the observed frequency change due to the Doppler effect is derived, and examples of its application are provided, such as the change in pitch of a siren as an ambulance passes by.

25:09

🎵 Wave Interference and the Doppler Effect in Sound Waves

Building on the previous discussions, the professor examines the interference of waves, particularly the interference pattern created by sound waves. He explains how the Doppler effect influences the interference pattern when either the source or the observer is in motion. The lecture also touches on the relativistic effects on the Doppler shift for light and the phenomenon of sonic booms, which occur when an object travels faster than the speed of sound, causing a buildup of compressed sound waves.

30:11

🌐 Linearity of Wave Equations and Wave Superposition

The lecture continues with an exploration of the linearity of wave equations, which allows for the superposition of waves. The professor demonstrates that the sum of two wave solutions is also a solution, leading to a discussion of wave interference. He uses the example of two sound waves with slightly different frequencies to illustrate how they can constructively or destructively interfere, resulting in a phenomenon known as beats. The mathematical expression for the resulting wave from two interfering waves is derived, and the conditions for constructive and destructive interference are explained.

35:13

🎼 Application of Interference in Musical Instruments and Physics

The professor discusses the practical applications of wave interference, such as tuning musical instruments using beat frequencies. He explains how radio stations use carrier waves and modulation to create audible beats. The lecture then transitions to a more complex example of wave interference: the double-slit experiment. The professor describes how waves passing through two slits create an interference pattern, with points of constructive and destructive interference depending on the path difference between the waves from each slit.

40:14

📏 Conditions for Constructive and Destructive Interference

The lecture delves into the specific conditions that lead to constructive and destructive interference in the context of the double-slit experiment. The professor explains how to calculate the positions of these interference patterns using the wavelength of the waves and the distance between the slits. He introduces the concept of path difference and how it relates to the angles at which constructive and destructive interference occur. The professor also discusses an approximation method for determining these angles when the observation screen is far from the slits.

45:14

🎶 Vibrations of a String Fixed at Both Ends

The professor explores the vibrations of a string fixed at both ends, which can only occur at specific frequencies corresponding to standing wave patterns. These patterns include nodes (points of no vibration) and anti-nodes (points of maximum vibration). The lecture explains that the fundamental frequency and its integer multiples are the only frequencies at which such a string can vibrate. The relationship between the length of the string, the wavelength, and the frequency is discussed, with examples of how to calculate the frequencies of the allowed modes of vibration.

50:16

🎵 Frequencies of Air Columns in Tubes

The lecture concludes with an examination of the frequencies of air columns in tubes, which behave similarly to strings fixed at both ends. The professor describes how the modes of vibration for air in a tube are determined by the presence of nodes at the closed ends and anti-nodes in between. He explains the relationship between the length of the tube, the wavelength, and the frequency for different scenarios, such as tubes closed at both ends, open at both ends, and closed at one end and open at the other. The lecture highlights that the frequencies for tubes with both ends the same (open or closed) are integer multiples of the fundamental frequency, while tubes with one open and one closed end have frequencies that are odd multiples of the fundamental frequency.

Mindmap

Keywords

💡Wave Equation

The wave equation is a fundamental differential equation that describes the propagation of oscillations in physical media. In the context of the video, it is used to explain how waves travel along a string, with the equation d^(2)ψ/dx^(2) = 1/v^(2) d^(2)ψ/dt^(2), where ψ represents the displacement of the string at a point x and time t, v is the wave speed, and F/μ (tension divided by mass per unit length) replaces v^2. This equation is central to understanding wave behavior on a string and is a key theme in the video.

💡String Vibrations

String vibrations refer to the oscillations or waves produced when a string is plucked or disturbed from its equilibrium position. The video discusses string vibrations as a concrete example of wave motion, using the displacement ψ(x, t) to describe the motion at any point x on the string and time t. The concept is integral to the discussion of wave physics and the derivation of the wave equation.

💡Tension

Tension, in the context of the video, is the force that acts along the string, pulling it taut. It is a crucial factor in wave propagation on the string, as it, along with the mass per unit length (μ), determines the wave speed (v). The video explains that v^2 is equal to the tension (F) divided by μ, which is a fundamental relationship in string wave dynamics.

💡Displacement

Displacement in the video refers to the transverse movement of the string from its equilibrium position due to wave motion. It is represented by the variable ψ(x, t), which denotes the height of the string above its normal position at a point x and time t. Displacement is a key concept in understanding how waves manifest on a string.

💡Curvature

Curvature, as discussed in the video, is related to the second derivative of the displacement function, which describes the bending or the rate of change of the slope of the string's wave. A positive curvature indicates the string is below its normal position and experiences an upward force, while a negative curvature suggests the string is above the normal position and experiences a downward force. Curvature is essential in deriving the net force acting on an element of the string.

💡Wavelength

Wavelength (λ) is the distance between two consecutive points on a wave that are in phase with each other, such as from one crest to the next. In the video, the relationship between wavelength and the constant k is established as kλ = 2π, which is a fundamental aspect of wave properties. Wavelength is used to describe the periodic nature of waves on a string and is key to understanding wave patterns.

💡Frequency

Frequency in the video is the number of oscillations or cycles a wave completes in a given time period and is represented by the Greek letter ω (omega). It is related to the time period (T) of a wave as ω = 2π/T. Frequency is a critical concept in wave analysis, as it, along with wavelength, determines the properties of the wave, such as its speed.

💡Beats

Beats are a perceptible fluctuation in amplitude caused by the interference of two waves with nearly the same frequency. In the video, beats are produced when two sources of sound with slightly different frequencies (ω1 and ω2) are combined, resulting in a periodic increase and decrease in amplitude. The beat frequency is the difference between the two frequencies, which is an important concept in tuning musical instruments and understanding wave interference.

💡Interference

Interference in the video refers to the phenomenon where two or more waves superimpose to form a resultant wave of greater, lower, or the same amplitude. The video discusses both constructive interference, where waves add up to increase amplitude, and destructive interference, where they cancel each other out. This concept is crucial in explaining phenomena such as the patterns formed in the double-slit experiment and noise cancellation.

💡Doppler Effect

The Doppler effect is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. In the video, it is explained that when a source moves towards an observer, the frequency appears to increase, and when it moves away, the frequency decreases. This effect is fundamental to understanding wave behavior in various contexts, such as the change in pitch of a siren as it approaches or recedes from an observer.

💡Energy Density

Energy density in the video is the energy stored in a wave per unit length of the medium. It is calculated as the sum of kinetic and potential energy densities. The video explains that for a vibrating string, the energy density (u) is given by ½μA^2ω^2, where μ is the mass per unit length, A is the amplitude, and ω is the angular frequency. This concept is essential for understanding the energy transport properties of waves.

💡Sound Intensity

Sound intensity is the power carried by sound waves per unit area in a given direction. In the video, it is discussed in the context of how the intensity of sound waves decreases with distance from the source in three-dimensional space, following the inverse square law (P = I_0 / 4πr^2), where P is the power, I_0 is the standard intensity, and r is the distance from the source. Intensity is a key concept in acoustics and the perception of loudness.

💡Decibels

Decibels (dB) are a unit used to express the level of sound intensity in relation to a reference level, which is set at 10^-12 watts per meter squared. In the video, decibels are explained as a logarithmic measure of the ratio of a given intensity to the standard intensity, multiplied by 10. This scale is used because the human ear can perceive a wide range of sound intensities, and the logarithmic scale makes it easier to represent and compare these differences.

Highlights

Introduction to wave theory using a string as a medium to illustrate wave behavior.

Derivation of the wave equation for a string under tension, relating tension, mass per unit length, and wave speed.

Explanation of the wave equation's relation to Newton's second law (F=ma) for a small segment of the string.

Discussion on the small angle approximation and its implications for the horizontal and vertical forces in the wave equation.

Presentation of the general solution to the wave equation, A cos(kx - ωt), and the condition ω = kv for the solution to hold.

Clarification on the limitations of the solution's amplitude A due to the small displacement approximation.

Demonstration that any function of (x - vt) satisfies the wave equation, highlighting wave properties.

Analysis of wave velocity and its relation to the wave's shape and movement over time.

Definition and calculation of wavelength (λ) and its relationship with wave number (k).

Connection between angular frequency (ω), time period (T), and their reciprocal relationship with wave properties.

Calculation of energy in a vibrating string and the concept of energy per unit length.

Introduction to the power required to maintain a wave and its relation to energy per unit length and wave velocity.

Explanation of intensity in three dimensions and the concept of decibels as a measure of sound intensity.

Discussion on the Doppler effect and the change in perceived frequency due to the relative motion of source and observer.

Application of wave theory to sound, including the Doppler effect and sonic booms.

Introduction to wave interference, a key property of waves, and its implications for sound and vibration.

Detailed exploration of the double-slit experiment with waves, leading to constructive and destructive interference patterns.

Practical applications of wave interference in noise cancellation and musical instruments.

Analysis of standing waves on a string fixed at both ends and the calculation of possible vibrational modes and frequencies.

Discussion on the vibrational modes of air columns in tubes, both open and closed, and their respective frequencies.

Conclusion on the fundamental principles of wave behavior as applied to strings and tubes, emphasizing the importance of understanding wave properties.

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Wave Physics Lecture Notes Sound Waves String Vibrations Wave Equations Doppler Effect Interference Patterns Beat Frequencies Acoustic Properties Educational Content