Waves

Types of waves

The Great Wave off Kanagawa, Hokusai, 1829

Wave phenomena are ubiquitous in the natural world. Nearly every discipline in science must be able to deal with waves or wave-like behavior. Consider a geologist studying earthquakes. The motions of the earth following tectonic activity are waves. Understanding how they move through the earth's surface is paramount to dealing with these disasters. If you're an electrical engineer, you'll need to understand electromagnetic waves thoroughly. Even economists can analyze certain properties of the global economy in terms of wave physics.

Perhaps the most important feature of all waves, is that they do not move matter in the direction of travel, only energy is transmitted. We'll see how this makes sense as we consider the various types of waves.

Mechanical Waves

These waves require a medium. A disturbance in the medium propagates through the medium and this is called the wave.

Water waves
Sound waves
Stadium waves (excited people)

Electromagnetic Waves

These are oscillations of electromagnetic fields. These waves don't require a medium.

Light
Radio waves
wi-fi

Matter Waves

Very small particles have wave-like properties. (We'll have to save that for later.)

Mechanical Waves

These are the most familiar.

They require a medium.

A wave pulse

A sinusoidal wave

Two types of mechanical waves

Transverse and Longitudinal Waves

There are 2 general classes of waves:

Longitudinal: The Particles move parallel to the direction in which the wave travels.

Transverse: Particles in the medium move perpendicular to the direction that wave is traveling

Imagine one small segment of the rope. Let's look at its motion while the wave is passing through that point.

We'll need to be able to quantify the motion of this little blue region.

Waves, a mathematical formulation

Imagine a small segment of the rope, or spring, or water or whatever medium. To fully describe its motion, we'll need variables which tell us where it is and when -- essentially space and time. If we imagine a rope, which we can say only has two spatial dimensions, then we just need three total variables: $x,y,$ and $t$.

We can therefore completely describe the motion of a segment using: $$y = h(x,t),$$ where $h$ is a function of space and time.

$y$ is simply the transverse displacement of a small element of the string.

A basic wave

Here's a sinusoidal wave traveling in the $+x$ direction.

The dot shows one element of the medium oscillating up and down, sinusoidally.

So, let's use a sine function to describe the position of a small section of the medium.

$$\bbox[10px,border:2px solid red]{y(x,t) = A \sin (kx - \omega t)}$$

Now, we have 3 variables in one function. This means that in order to plot this on a 2D graphs, we'll need to keep one variable constant.

Depending on which one, we'll end up with two different representations of the same wave.

Amplitude

Here we refer to the maximum displacement of the elements away from their equilibrium positions.

This plot shows two waves with amplitudes which differ by a factor of two.

Wavelength

Let's take a sine wave at time $t = 0$. Our previous formula will then be:

$$y(x,0) = A \sin (kx)$$

Now, the displacement is the same at both ends of the wavelength

$$ \begin{eqnarray} A \sin (kx_1) & = & A \sin k(x_1+\lambda)\\ & = & A \sin (kx_1 + k \lambda) \end{eqnarray} $$

This can only be true if $$\bbox[10px,border:2px solid red]{k = \frac{2\pi}{\lambda}}$$

We'll call $k$ the wavenumber, which we can see is inverse to the wavelength.

Period and Frequency

$$\bbox[10px,border:2px solid red]{f = \frac{1}{T} = \frac{\omega}{2 \pi}}$$.

What should the horizontal axis be labeled as?

Position
Time
Amplitude
Something else

Vectors are attached to several particles on this wave. What vectors are shown?

Displacement
Velocity
Acceleration
Force

The velocity of a segment.

$$y(x,t) = A \sin (kx - \omega t)$$

Here is our position equation for the material of the wave. If we wanted to find the velocity of a given point, we would just need to take the time derivative of this equation.

Here is a snapshot of a traveling wave on a rope. It's moving in the $-x$ direction. Which of the labeled segments has the largest negative y-velocity value?

E. All have zero y-velocity values.

Here is a snapshot of a traveling wave on a rope. It's moving in the $- x$ direction. Which of the labeled segments has the largest positive y-acceleration value?

E. All have zero y-acceleration values.

Here is a history graph for a segment of rope. A wave is traveling in the $+x$ direction and the rope is being deformed in the $y$ axis. What is the wavelength of the wave?

1 m
1.5 m
2 m
3 m
Cannot be determined from the information given

What is the frequency, $f$, of the wave?

.5 Hz
2/3 Hz
1.5 Hz
3 Hz
Cannot be determined from the information given

Speed of a traveling wave

If we think about the crest of the wave as it moves, its displacement in the $y$ axis is constant.

For that to be the case, the argument of the sinusoidal term must be constant as well.

$$y(x,t) = A \sin (\;\underbrace{kx - \omega t}_{\textrm{constant}}\;)$$

or, $kx - \omega t = \textrm{constant}$

Let's take the time derivative of that equation: $$k \frac{dx}{dt} - \omega = 0$$ -or- $$\frac{dx}{dt} = v = \frac{\omega}{k}$$

rewriting in more familiar terms:

$$\bbox[10px,border:2px solid red]{v = \lambda f}$$

Recall that the frequency, $f$, is just the inverse of the period, $T$. Using this, we can rearrange this expression: $$v = \lambda f = \frac{\lambda}{T} \Rightarrow \frac{\textrm{distance}}{\textrm{time}}$$ We should be somewhat comforted by the fact that this is our familiar distance over time expression. (wavelegth is a distance, period is a time. )

Note that this is just the velocity of the wave, not the velocity of the material in the wave. That velocity is given by the derivative of the position equation: $$\frac{dy}{dt}=\dot{y}(x,t) = v(x,t) = -A \omega \cos \left(kx - \omega t\right)$$

This is a critical distinction to make. The wave speed, sometimes also called $c$, does not change in time. $dc/dt = 0$, whereas the transverse velocity of a given segment most certainly does change. $$\frac{d\dot{y}}{dt}= \ddot{y} \neq 0$$.

Wave speed on a real string.

What determines the speed of a traveling wave on a stretched string?

We could show by looking at the tensions in the string that the velocity must be determined by the tension, $\tau$, and the linear density, $\mu$. $$\bbox[10px,border:2px solid red]{v = \sqrt{\frac{\tau}{\mu}}}$$

Derive the wavespeed

Which of the following actions would make a pulse travel faster down a stretched string?

Use a heavier string of the same length, under the same tension.
Use a lighter string of the same length, under the same tension.
Move your hand up and down more quickly as you generate the pulse.
Move your hand up and down a larger distance as you generate the pulse.
Use a longer string of the same thickness, density, and tension.

A thick heavy (i.e. not massless)rope is hanging from a very tall ceiling. A person grabs the end of the rope and begins moving it back and forth with a constant amplitude and frequency. A transverse wave moves up the rope. Which of the following statements describing the speed of the wave is true?

The speed of the wave decreases as it moves upward.
The speed of the wave increases as it moves upward.
The speed of the wave is constant as it moves upward.
The speed of the wave does not depend on the mass of the rope.
The speed of the wave depends on its amplitude.

A wave travels along a string at speed $v_0$ . What will the speed be if the string is replaced by one made of the exact same material, but having twice the radius. (The tension is the same)

A sinusoidal wave with an amplitude of 1.00 cm and a frequency of 100 Hz travels at 200 m/s in the positive x-direction. At t=0s, the point x = 1.00 m is on a crest of the wave.

Determine $A, v, \lambda, k, f, \omega, T, \textrm{ and } \phi$ for this wave.
Write the equation for the wave's displacement as it travels.

Energy

An interesting aspect of waves is that they are a means of transferring energy, but not matter.

Here is a frame from the earlier animation. The vectors show the velocity of each element of mass, $dm$. At the crest, ②, the velocity is zero. While, at the $y = 0$ point, ①, the velocity will be a maximum.

Kinetic energy is given by the square of the velocity, thus we can see that the K.E. of the $dm$ element will oscillate between a minimum and maximum during the wave travel.

Energy, cont'd

We should also consider the potential energy of the element $dm$.

At point ②, the string is not stretched at all, while at point ①, the string length is elongated as it passes through the origin. This change in length will change the elastic potential energy.

Quantify energy

The kinetic energy of a particle in motion is given by $KE = \frac{1}{2}m v^2$. In the case of a little section of a rope, $dm$, we write: $$dK = \frac{1}{2}dm \dot{y}^2$$ ($\dot{y} = \frac{dy}{dt}$ which is the time derivative of $y$, or speed in the transverse direction)

Going back to our original definition for the $y$ displacement: $y(x,t) = A \sin\left(kx - \omega t\right)$, we can see that $\dot{y}$ is just: $$\dot{y} = -A\omega \cos\left(kx - \omega t\right)$$

Therefore, our dK can be written: $$dK = \frac{1}{2} dm (-A\omega)^2\cos^2\left(kx - \omega t\right) = \frac{1}{2} \mu dx (-A\omega)^2\cos^2\left(kx - \omega t\right) $$

Quantify energy

If we continue, and take the derivative of dK with respect to time: $$\frac{dK}{dt} = \frac{1}{2} \mu v A^2\omega^2\cos^2\left(kx - \omega t\right)$$

Now, the kinetic energy of one little element is clearly changing all the time, but we can consider the average change in kinetic energy: $$\left(\frac{dK}{dt}\right)_\textrm{avg} = \frac{1}{2} \mu v A^2\omega^2\left[\cos^2\left(kx - \omega t\right)\right]_\textrm{avg}$$

The last term: $\left[\cos^2\left(kx - \omega t\right)\right]_\textrm{avg}$ is equal to $\frac{1}{2}$, so in the end, for the average rate of kinetic change: $$\left(\frac{dK}{dt}\right)_\textrm{avg} = \frac{1}{4} \mu v A^2\omega^2$$

The wave also transmits elastic potential energy (since the rope is kinda springy). This should be equal to the average kinetic energy since they are conserved quantities. Thus, the total rate of energy transmission (aka Power) will be twice what we figured for the kinetic energy: $$\bbox[10px,border:2px solid red]{ P_\textrm{avg}= \frac{1}{2} \mu v A^2\omega^2 }$$

The wave equation

This: $y(x,t) = A \sin (kx - \omega t)$ was just a specific case of a wave. (A sinusoidal traveling wave). We'll need a more general wave equation which can be used to describe any travelling wave.

**Derive** $$\bbox[10px,border:2px solid red]{ \frac{\partial^2 y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}}$$ Derive the wave equation

Sound waves

A bell is ringing inside of a sealed glass jar that is connected to a vacuum pump. Initially, the jar is filled with air at atmospheric pressure. What does one hear as the air is slowly removed from the jar by the pump?

The sound intensity gradually increases.
The sound intensity gradually decreases.
The sound intensity of the bell does not change.
The frequency of the sound gradually increases.
The frequency of the sound gradually decreases.

Speed of Sound

We saw before that the speed of a wave (transverse) was equal to: $$v = \sqrt{\frac{\tau}{\mu}}$$ In this case, $\tau$ was a type of 'elastic property' while $\mu$ would be classified as an 'inertial property'. i.e. $$ \sqrt{\frac{\textrm{elastic proptery}}{\textrm{inertial property}}} = v$$ These two quantities only made sense in reference to a string, but the speed of other types of waves can be determined by analogous considerations.

Bulk Modulus

$$\Delta p = - B \frac{\Delta V}{V}$$

$B$ is the Bulk modulus of a material. It tells us how the volume of given material will change if pressure is applied to it. Steel for example, has a $B = 1.6 \times 10^{11} $N/m$^2$, while water, which is a bit more compressible, has a $B = 2.2 \times 10^{9} $N/m$^2$

We'll use this parameter to calculate how compression waves travel in a medium (i.e. sound waves)

Material	v (m/s)
Gases
Hydrogen (0°C)	1286
Helium (0°C)	972
Air (20°C)	343
Air (0°C)	331
Liquids
Sea water	1533
Water	1493
Solids
Diamond	12000
Pyrex glass	5640
Iron	5130
Aluminum	5100
Copper	3560
Gold	3240
Rubber	1600

Here is a table that lists the velocity of sound in various materials.

The speed of sound is dictated by the material properties of the medium. Just like it was for the string.

$$v_\textrm{sound} = \sqrt{\frac{B}{\rho}}$$

A hammer taps on the end of a 4.00 m long metal bar at room temperature. A microphone at the other end of the bar picks up two pulses of sound, one that travels through the metal and one that travels through the air. The pulses are separated in time by 9.00 ms. What is the speed of sound in this metal?

You are observing a thunderstorm. In the distance, you see a flash of lightning. Five seconds later, you hear thunder. How far away was the lightning flash?

1 mile (1.6 km)
.5 mile (.8 km)
2 miles (3.2 km)
.25 miles (.4 km)
5 miles (8.0 km)

Traveling Sound Waves

To describe the longitudinal motion of an element of the medium, we can use a sinusoidal function:

$$s(x,t) = s_m \cos (kx - \omega t)$$

All of our wave parameters are still present: $f, \lambda, \omega, k, T$

However, it's easier to work with pressure, $p$.

The volume of our element will just be the length times its cross-sectional area: $$V = A\Delta x. $$ While the change in volume of this element will be given by: $$\Delta V = A \Delta s.$$

We can substitute these back into our formula for the bulk modulus: $$\Delta p = -B\frac{\Delta s}{\Delta x} = -B\frac{\partial s}{\partial x}$$

$$\frac{\partial s}{\partial x} = -ks_m \sin(kx - \omega t)$$ or $$\Delta p = B k s_m \sin(kx - \omega t)$$

Thus, we can see the pressure at a given location oscillates with time.

Graphical Representation of sound waves

We'll see plots like this a lot. Here we have a speaker producing a sound wave. We'll plot the displacement of the elements of air as a function of position.

A particle of dust is floating in the air approximately one half meter in front of a speaker. The speaker is then turned on produces a constant pure tone of 267 Hz, as shown. The sound waves produced by the speaker travel horizontally. Which one of the following statements correctly describes the subsequent motion of the dust particle?

The particle of dust will oscillate in the $\pm x$ direction with a frequency of 267 Hz.
The particle of dust will oscillate in the $\pm y$ with a frequency of 267 Hz.
The particle of dust will be accelerated toward the right ($+x$) and continue moving in that direction.
The particle of dust will move toward the right (+$x$) at constant velocity.
The dust particle will remain motionless as it cannot be affected by sound waves.

Radially Propagating Waves

Here is our standard, circular wave pattern.

We can approximate the circular waves as parallel wave fronts if we are far enough away from the source. (far would mean $d \gg \lambda$).

Wave fronts

We'll consider an intensity, $I$, to describe and quantify the loudness of the waves. $$I = \frac{P}{A}$$ Here, $P$, is the rate of energy transfer (power) and $A$ is the area over which we are considering.

To find the power at some distance, $r$, from the source, we need to use the surface area of a sphere at that distance: $$I = \frac{P}{A} = \frac{P}{4 \pi r^2}$$

A laser pointer emits 1 mW of light power into a 1.0 mm diameter laser beam. What is the intensity of the laser beam?

Nancy is a distance d in front of a speaker emitting sound waves. She then moves to a position that is a distance 2d in front of the speaker. By what percentage does the sound intensity decrease for Nancy between the two positions?

10%
25%
50%
75%
The sound intensity remains constant because it is not dependent on the distance.

Intensity

To quantify how loud a given sound is, we'll define a term $\beta$: the sound intensity level. It is calculated by comparing the intensity of the sound in question, $I$, to a base level, the threshold of hearing: $I_0$.

$$\beta = (10 dB) \log \frac{I}{I_0}$$

The threshold of hearing $I_0$ defined as $I_0 = 1.0 \times 10^{-12} \;\rm W/m^2$

The units of $\beta$ are given in decibels.

If a sound has an intensity of $1.0 \times 10^{-12} \;\rm W/m^2$, then we can see that it will have an intensity level of 0 dB.

$$\beta = (10 \textrm{dB})\log_{10}\frac{I_0}{I_0} = (10 \textrm{dB}) \log_{10}(1) = 0 \; \rm dB$$

A very loud sound, one that might damage your ears, can have an intensity of 10 W/m$^2$. How many decibels is that?

Sound	$\beta$ (dB)	$I$ W/m$^2$
Threshold of Hearing	0	$1.0 \times 10^{-12}$
A Whisper at 1m	20	$1.0 \times 10^{-10}$
Conversation at 1m	60	$1.0 \times 10^{-6}$
Vacuum Cleaner	80	$1.0 \times 10^{-4}$
Home Stereo	110	0.1
Threshold of PAIN	130	10

We can see that there is wiiiide range of intensities heard during a normal day out and about. That is why it makes more sense to use a log scale when describing this phenomenon.

If the sound intensity level at distance d of one trombone is $\beta$ = 70 dB, what is the sound intensity level of 99 identical trombones, all at distance $d$?

The Grateful Dead and their crew built a speaker system that was able to generate 26,400 Watts of audio power. It was called the Wall of Sound. How many decibels would this make at ¼ mile away from the stage? (Assume an isotropic sound distributioin)

Doppler

General Doppler Shift

$$f' = f \frac{v \pm v_D}{v \pm v_S}$$

$f$ = emitted frequency
$f'$ = detected frequency
$v$ = speed of sound in air
$v_D$ = speed of detector
$v_S$ = speed of source

When the motion of detector or source is toward the other, the sign on its speed must give an upward shift in frequency. When the motion of detector or source is away from the other, the sign on its speed must give a downward shift in frequency.

A child is swinging back and forth with a constant period and amplitude. Somewhere in front of the child, a stationary horn is emitting a constant tone of frequency $f$. Five points are labeled in the drawing to indicate positions along the arc as the child swings. At which position(s) will the child hear the lowest frequency for the sound from the horn?

at 2 when moving toward 1
at 2 when moving toward 3
at 3 when moving toward 2
at 3 when moving toward 4
at both 1 and 4

A 2kHz sine wave generator is swung around in a circle with a rope of length 1m at a speed of 100 rotations per minute. Find the highest and lowest frequencies heard by stationary listeners out side the circle but in the plane of rotation.