Chapter 15 OutlineMechanical 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

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.

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

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.

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.

Transverse Sinusoidal Waves

• As we discussed last chapter, SHM is described by its amplitude, , and period, , (as well as the corresponding frequency, , and angular frequency, .

• Over one period, the wave advances a distance we call the wavelength, .

• Distance for one full wave pattern.

• The wave speed (distance divided by time) is therefore .

• It is more commonly written in terms of the frequency.

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).

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.

• Consider the case of a sinusoidal wave with amplitude and angular frequency , propagating at speed .

Wave Function

• So, the wave can be described by:

• The wave number, , is defined as:

• Since , the wave velocity is:

Note: This is referred to as the phase velocity.

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,

• Taking partial derivatives with respect to position and time, we find that

• This is called the wave equation, and it is true of any periodic wave, even non-mechanical waves (light).

Speed of a Wave on a String

• Consider a small segment of the string, length , mass , with a linear mass density .

• Normally we use lambda for linear density, but we are using it already for wavelength.

• The forces acting on either end of the segment, and , 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.

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.

Speed of a Wave on a String

• After some manipulation,

• The wave equation:

• So, the wave speed must be:

• Since is the horizontal component of the force it must be equal to the tension, .

• We are not using so as not to confuse it with the period.

Wave on a String Example

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 () can be expressed as

• 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.

• Combining these,

Energy in Wave Motion

• For a sinusoidal wave,

• So the power is

• Combining this with and

Maximum and Average Power

• Since sine of any angle cannot exceed one, the maximum power is:

• The average power is then

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, , to be the average power transported per unit area.

• This means that the intensity decreases with .

Wave Power Example

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.

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.

Standing Waves on a String

• Consider a sinusoidal wave on a string.

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

Mathematical Description of Standing Wave

• A standing wave is the superposition of two waves traveling in opposite directions.

• Equivalently, a wave and its reflection

• Using the identity ,

• So, the string has a shape given by the term, and its amplitude oscillates with the term.

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, .

• The frequencies are called the harmonics.

• Remember that the distance between adjacent nodes is .

Normal Modes of a String

• Call the length of the string .• In the mode, there are

antinodes.

• So, the wavelength of the mode is:

• Since and the wave speed is constant,

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.

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).

Harmonics on a String

• Consider a string tuned to A2 ().• The note is named after the fundamental frequency.

• The higher harmonics are given by .

• 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, ?

• Looking at the table, it is very close to an E.

• What about the third harmonic, ?• While it is not as exact, this is quite close to an C#.

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.

Chapter 15 SummaryMechanical Waves

• Mechanical waves• Transverse or longitudinal

• Transport energy, not matter

• Periodic waves• Wave number:

• Angular frequency:

• Wave speed:

• Period:

• Amplitude:

• Wave equation:

• Speed of wave on a string:

Chapter 15 SummaryMechanical Waves

• Energy and intensity

• (for 3D)

• Superposition of waves

• Standing waves on strings: • Fundamental frequency:

• Harmonics: ;

• Stringed instruments