1. Chapter 15 Outline Mechanical Waves
Mechanical wave types
Periodic waves
Sinusoidal wave
Wave equation
Speed of wave on a string
Energy and intensity
Superposition of waves
Standing waves on strings
Normal modes
Stringed instruments

2. Mechanical Wave
A disturbance that travels through some medium is a mechanical wave.
Sound, ocean waves, vibrations on strings, seismic waves…
In each case, the medium moves from and returns to its equilibrium state.
The wave transports energy, not matter.
The disturbance propagates with a wave speed that is not the same as the speed at which the medium moves.

3. Transverse Wave
If the motion of the medium is perpendicular to the propagation of the wave, it is a transverse wave.
e.g. wave on a string

4. Longitudinal Wave
If the motion of the medium is along the propagation of the wave, it is a longitudinal wave.
e.g. sound waves
We can also have a combination of the two.

5. Periodic Waves
If we shake a stretched string once, a wave pulse will travel along the string, but afterwards the string returns to its flat equilibrium position.
If instead, we attach the string to a simple harmonic oscillator, we will produce a periodic wave.
A simple example is a sinusoidal wave.

6. Transverse Sinusoidal Waves
As we discussed last chapter, SHM is described by its amplitude, 𝐴, and period, 𝑇, (as well as the corresponding frequency, 𝑓=1/𝑇, and angular frequency, 𝜔=2𝜋𝑓.
Over one period, the wave advances a distance we call the wavelength, 𝜆.
Distance for one full wave pattern.
The wave speed (distance divided by time) is therefore 𝑣=𝜆/𝑇.
It is more commonly written in terms of the frequency.
𝑣=𝜆𝑓

7. Longitudinal Sinusoidal Waves
In a longitudinal wave, the medium oscillates along the direction of propagation.
A common example is a sound wave.
The wave is composed of compressions (high density) and rarefactions (low density).

8. Wave Function
As in the harmonic motion from last chapter, we want to mathematically describe the displacement of the medium.
Since the wave is moving, the displacement depends on both the position and the time.
𝑦=𝑦(𝑥,𝑡)
Consider the case of a sinusoidal wave with amplitude 𝐴 and angular frequency 𝜔, propagating at speed 𝑣.

9. Wave Function So, the wave can be described by: 𝑦 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡)
The wave number, 𝑘, is defined as:
𝑘= 2𝜋 𝜆
Since 𝜔=2𝜋𝑓, the wave velocity is:
𝑣=𝜆𝑓= 2𝜋 𝑘 𝜔 2𝜋
𝑣= 𝜔 𝑘
Note: This is referred to as the phase velocity.

10. The Wave Equation
We have described sinusoidal waves, but we can derive an expression that will hold for any periodic wave.
Starting with the sinusoidal wave function,
𝑦 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡)
Taking partial derivatives with respect to position and time, we find that
𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2
This is called the wave equation, and it is true of any periodic wave, even non-mechanical waves (light).

11. Speed of a Wave on a String
Consider a small segment of the string, length ∆𝑥, mass 𝑚, with a linear mass density 𝜇=𝑚/∆𝑥.
Normally we use lambda for linear density, but we are using it already for wavelength.
The forces acting on either end of the segment, 𝑭 1 and 𝑭 2 , pull along the string. (At rest, these would just be the tension, and would cancel each other.)
Because the string is displaced from equilibrium, these forces are not parallel.
The horizontal components cancel, but the vertical components lead to the transverse acceleration.

12. Speed of a Wave on a String
By comparing the forces to the slope of the string at that point, we can relate the horizontal force to the transverse acceleration.
Using the wave equation, we can then find the wave speed.

13. Speed of a Wave on a String
After some manipulation,
𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 𝜇 𝐹 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2
The wave equation:
𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2
So, the wave speed must be:
𝑣= 𝐹 𝜇
Since 𝐹 is the horizontal component of the force it must be equal to the tension, 𝑇.
We are not using 𝑇 so as not to confuse it with the period.

14. Wave on a String Example

15. Energy in Wave Motion
As we discussed at the beginning of class, waves do not transport matter, but they do transport energy.
Recall that power (𝑃=𝑑𝑊/𝑑𝑡) can be expressed as
𝑃=𝐹𝑣
Consider a small segment of the string. It moves up and down because the wave is doing work on it.
Only the vertical component of the force does work, and the velocity is also vertical.
𝐹 𝑦 =−𝐹 𝜕𝑦 𝜕𝑥
𝑣 𝑦 = 𝜕𝑦 𝜕𝑡
Combining these,
𝑃=−𝐹 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑡

16. Energy in Wave Motion For a sinusoidal wave, 𝑦 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡)
So the power is
𝑃=−𝐹 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑡 =−𝐹[−𝑘𝐴 sin 𝑘𝑥−𝜔𝑡 ][𝜔𝐴 sin 𝑘𝑥−𝜔𝑡 ]
𝑃=𝐹𝑘𝜔 𝐴 2 sin 2 𝑘𝑥−𝜔𝑡
Combining this with 𝑘=𝜔/𝑣 and 𝑣= 𝐹/𝜇
𝑃= 𝜇𝐹 𝜔 2 𝐴 2 sin 2 𝑘𝑥−𝜔𝑡

17. Maximum and Average Power
Since sine of any angle cannot exceed one, the maximum power is:
𝑃 max = 𝜇𝐹 𝜔 2 𝐴 2
The average power is then
𝑃 ave = 𝜇𝐹 𝜔 2 𝐴 2

18. Wave Intensity
A wave on a string only transports energy in one dimension, but many other waves propagate in three dimensions.
If this propagation is uniform, the power is spread out over the surface of a sphere, and we define the intensity, 𝐼, to be the average power transported per unit area.
𝐼= 𝑃 𝐴 =𝑃/4𝜋 𝑟 2
This means that the intensity decreases with 1/ 𝑟 2 .
𝐼 1 𝐼 2 = 𝑟 𝑟 1 2

19. Wave Power Example

20. Boundary Conditions and Reflection
When a wave propagates along a string, what happens when it reaches the end?
It will reflect from the end.
This reflection depends on the boundary conditions, whether the end is free or fixed.

21. Superposition
When waves overlap, the net displacement is simply the sum of the displacements of each wave.
This is called superposition.
If the displacements are equal but opposite, they cancel each other, and we have destructive interference.
If the displacements have the same sign, they add together, and we have constructive interference.

22. Standing Waves on a String
Consider a sinusoidal wave on a string.

23. Standing Waves on a String
Points at which the waves always interfere destructively and the string does not move are nodes.
Points of the greatest amplitude correspond to constructive interference and are called antinodes.
Distance between any two nodes or any two antinodes is always one half the wavelength

24. Mathematical Description of Standing Wave
A standing wave is the superposition of two waves traveling in opposite directions.
Equivalently, a wave and its reflection
𝑦 1 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡)
𝑦 2 𝑥,𝑡 =−𝐴 cos (𝑘𝑥+𝜔𝑡)
𝑦 𝑥,𝑡 = 𝑦 1 𝑥,𝑡 + 𝑦 2 𝑥,𝑡 =𝐴[ cos 𝑘𝑥−𝜔𝑡 − cos 𝑘𝑥+𝜔𝑡 ]
Using the identity cos (𝛼±𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽 ,
𝑦 𝑥,𝑡 =2𝐴 sin (𝑘𝑥) sin (𝜔𝑡)
So, the string has a shape given by the sin (𝑘𝑥) term, and its amplitude oscillates with the sin (𝜔𝑡) term.

25. Normal Modes of a String
If both ends of the string are fixed the standing wave must have nodes at each end.
This restricts the possible wave patterns that can occur on the string.
These are called the normal modes.
The mode with the longest wavelength oscillates at the fundamental frequency, 𝑓 1 .
The frequencies are called the harmonics.
Remember that the distance between adjacent nodes is 𝜆 2 .

26. Normal Modes of a String
Call the length of the string 𝐿.
In the 𝑛 th mode, there are 𝑛 antinodes.
𝐿=𝑛 𝜆 2
So, the wavelength of the 𝑛 th mode is:
𝜆 𝑛 = 2𝐿 𝑛
Since 𝑣=𝜆𝑓 and the wave speed is constant,
𝑓 𝑛 =𝑛 𝑣 2𝐿 =𝑛 𝑓 1

27. Timbre
A guitar, a harp, and a piano all produce music from standing waves on strings, so why do they sound different, even if they are playing the same note?
Timbre describes the tonal quality of a sound.
Envelope and spectrum.
The relative strength of the harmonics determines a large part of the distinctive sounds of different instruments.
This can be analyzed in more detail using a Fourier transform.

28. Basic Music Theory
If you sit at a piano and randomly hit keys, you find that some notes seem to go together while other combinations do not.
Can this be related to the vibrational modes of a single string?
Groups of notes played together are called chords, and one of the most basic harmonious type is the major chord.
It consists of the root, a major third, and a perfect fifth.
For example, an C major is a C, E, and G.
Notes are named by the letters A-G (with sharps/flats between all but B/C and E/F).

29. Harmonics on a String Consider a string tuned to A2 (110 Hz).
The note is named after the fundamental frequency.
The higher harmonics are given by 𝑓 𝑛 =𝑛 𝑓 1 .
𝑓=220 Hz, 330 Hz, 440 Hz, 550 Hz…
Anytime the frequency of a note is doubled, it is the same note an octave higher.
So, the second and fourth harmonics are also A.
What about the third harmonic, 330 Hz?
Looking at the table, it is very close to an E.
What about the third harmonic, 550 Hz?
While it is not as exact, this is quite close to an C#.

30. Basic Music Theory What do we get if we put an A, C#, and E together?
This is an A major chord.
The strongest harmonics in a single vibrating string make up a major chord.
Note that the frequencies were not exact.
In just intonation, the notes are defined such that the harmonic series perfectly replicates a major chord, but this can only work well for a single key.
Equal temperament is a compromise that works well across all keys.

31. Chapter 15 Summary Mechanical Waves
Transverse or longitudinal
Transport energy, not matter
Periodic waves
Wave number: 𝑘=2𝜋/𝜆
Angular frequency: 𝜔=2𝜋𝑓
Wave speed: 𝑣=𝜆𝑓=𝜔/𝑘
Period: 𝑇=1/𝑓
Amplitude: 𝐴
Wave equation: 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2
Speed of wave on a string: 𝑣= 𝐹 𝜇

32. Chapter 15 Summary Mechanical Waves
Energy and intensity
𝑃= 𝜇𝐹 𝜔 2 𝐴 2 sin 2 𝑘𝑥−𝜔𝑡
𝑃 ave = 𝜇𝐹 𝜔 2 𝐴 2
𝐼= 𝑃 𝐴 =𝑃/4𝜋 𝑟 2 (for 3D)
Superposition of waves
Standing waves on strings: 𝑦 𝑥,𝑡 =2𝐴 sin (𝑘𝑥) sin (𝜔𝑡)
Fundamental frequency: 𝑓 1 = 𝑣 2𝐿
Harmonics: 𝑓 𝑛 =𝑛 𝑣 2𝐿 =𝑛 𝑓 1 ; 𝜆 𝑛 = 2𝐿 𝑛
Stringed instruments