Waves and Sound NotesLevel 1: Wave Terminology and Characteristicsportions of this are from Knapp Notes

What Are Waves?Waves disturbances that travel through space transferring energy from one place to another.

Sound, light, and the ocean's surf are all examples of waves.

There are two key types of waves: mechanical (made of physical things, like water, sound, the earth…) and electromagnetic (light, x rays…).

Mechanical waves travel through a medium. The medium is whatever is being disturbed by the wave. If we are talking about a water wave, water is the medium. It’s important to note that the actual pieces of the medium don’t actually move with the wave. Think of The Wave in a stadium. The particles of the medium (people) move up and down. They don’t actual stay with the moving wave.

That waves carry energy should be obvious. Picture the waves on the ocean. Waves are generated far out at sea mainly by the wind. The wave travels through the water for hundreds or even thousands of miles. Finally it reaches the shore where the waves pound against the beach. They have enough energy to break down the coastline and erode away continents.

PulseA pulse is one single short wave.

Types of WavesThere are two types of mechanical waves, the transverse wave and the longitudinal wave.

Transverse Wave - The disturbance direction is perpendicular to the wave direction

Longitudinal Wave - The disturbance direction is parallel to the wave direction

D ist ur bance d ir ect ion

Wave directionTransver se Wave

L ongitudinal Wave

D ist ur bance d ir ect ion

Wave direction

There is also a third, less commonly discussed wave, called a circular wave. In circular waves, the particles move in a circle (go figure).

Take a moment to check out the animations that show each of these waves. It will really help clarify it for you.

Parts of a WaveYou are already familiar with some of these. Wavelength, frequency, amplitude…all of these terms make an appearance again here. Take a look at the wavelength in a longitudinal wave.

You’ve already dealt with frequency, but here is another way of thinking about it.

The frequency of a traveling wave is simply the number of cycles divided by the time they occur in.

Here f is the frequency, n is the number of cycles (and has no unit) and t is the time.

Practice ProblemA speed boat zooms by you as you lie on your floating mattress. You find yourself bobbing up and own on the waves that the boat made. So, you decide to do a little physics experiment. You count the waves and time how long it takes for them to go past. Six wave crests go by in five seconds. So what is the frequency?

Solution

Graphing WavesBelow is the plot of a transverse wave. The displacement is plotted on the y axis and distance is plotted on the x axis. The amplitude, A, is shown. This is the maximum displacement, just as it was for periodic motion. The other thing that is shown on the graph is the wavelength - . The wavelength is the distance between two in phase points on the wave.

The Wave EquationThe wave is traveling at some velocity v. We know that velocity is given by this equation:

We also know that the wave travels a distance of in the period, T.

We can plug these into the equation for velocity:

But we also know that the period is given by:

so

We can plug this in for T in the velocity equation we’ve been working on:

This gives us a very important equation:

This is called The Wave Equation Memorize This

Practice ProblemA middle C note (notes are these musical frequency kind of deals) has a frequency of approximately 262 Hz. Its wavelength is 1.31 meters. Find the speed of sound.Solution

Y

X

A

Practice ProblemA wave has a frequency of 25.0 Hz. Find the (a) wavelength, (b) period, (c) amplitude, and (d) velocity of wave. A graph of this wave is shown below.

Solution

(a) Amplitude: We can read the amplitude directly from the graph:

(b) Wavelength:This can be read directly from the graph as well.

(c) Period: The period is the inverse of the frequency, which we know.

(d) Velocity: We use the wave velocity equation.

Finding the Speed of a Wave in a MediumIn the last section, we looked at how you could calculate the wavelength/frequency if you know the speed of a wave in a medium.

Key Idea: The speed of a wave depends on the medium it is traveling through. If you have two different waves traveling through the same medium, they will have the same speed. It doesn’t matter if one has a higher amplitude or has a lower frequency. The medium is what determines the speed of the wave. The only way you can alter the speed of a wave is if you alter the medium.

There are several things that can affect the wave speed in a medium. We are not going to get into it deeply here, but you should know that this is at least in part determined by the molecular structure of the atoms in the material.

Temperature also affect wave speed. In general, the higher the temperature, the higher the wave speed. This makes sense when you think about it. The faster waves are moving, the more they will be able to hit off one another and move the wave forward.

One last thing- transverse and longitudinal waves will move through the same material at different speeds. The chart I used to explain above was for sound, which is a longitudinal wave. If I was curious about the speed that transverse waves move through a material, I would need a completely different chart with completely different values.

Supplemental: Earthquakes This is probably the most fascinating use of transverse and longitudinal waves. Awesome. This video explains it- but it is literally the cheesiest thing I have ever seen. I cannot watch this without laughing.

YX

35.0 cm

12.0 cm

There are lots of different equations that can help you calculate the speed of a wave in a medium. We are only going to look at one: how to calculate the speed of a wave moving along a string.

v=√ FTμWhere FT is the force of tension, v is the velocity and µ is the linear mass density. Linear mass density is the amount of mass per length in a string, or mass/length. A kite string, for example, would have a low linear mass density, while the rope holding a ship to a dock would likely have a high linear mass density.

So what is this equation saying? If we increase the tension in a string, the wave will move faster. If we choose a string with a higher linear mass density, the slower the wave will move. This makes sense right? If we have a fat, floppy rope, we won’t expect it to transmit a wave very quickly. If we have a lightweight, tight jump rope, we can probably get a wave moving pretty fast down it.

Practice Problem: Calculating with mass density

Solution

Practice Problem: Waves Entering a New Medium

Solution

Video Example: Twu Waves 3 You don’t have to watch this whole video if you don’t want to. What I do want you to do is start at 2:00, read the problem and try it yourself. Then watch her solve it and make sure you understand how to do these. Then, when she gets to part c, pause the video, try it yourself, then watch, learn. You get the idea.

Energy in a WaveWhat happens when a wave has traveled for a while and its starts to lose energy? Does it slow down?

No- because the material it is moving through doesn’t change. You can’t change the speed without changing the material. So what does change? The amplitude. The amplitude of a wave will decrease as the wave loses energy.

Video Lecture: Twu Waves 4 Great explanation of energy in waves. Definitely watch this. You can stop at 3:00.

Level 2: Wave DynamicsReflectionfrom Knapp Notes

When a wave traveling through a given medium encounters a new medium, two things happen: some of the energy the wave is carrying keeps going on into the new medium and some of the wave energy gets reflected back from whence it came. If the difference in the wave velocity is large, then most of the wave will be reflected. If the difference in velocity is small, most of the wave will be transmitted into the new medium. The junction of the two mediums is called a boundary.

If there is no relative motion between the two mediums, the frequency will not change on reflection. Also, and this is a key thing, the frequency does not change when the wave travels from one medium into another. It stays the same. This means that the wavelength does change.

There are two types of reflection. The type of reflection depends on how the mediums at the boundary are allowed to move. The two types are: fixed end reflection, and free end reflection.

For fixed end reflection think of the medium as being constrained in its motion. In the picture to the left you see a string that is securely fixed to the wall. The string (the old medium) is free to move up and down, but at the boundary where it meets the new medium (the wall) it is constrained – the string can’t really move up and down like it could before. In fixed end reflection, the wave that is reflected back is out of phase by 180. In the drawing you see an erect pulse traveling down the string. When it is reflected it ends up inverted. It will have the same speed going in as coming out. So in fixed end reflection an erect pulse would be reflected as an inverted pulse.

In free end reflection, the medium is free to move at the boundary. The reflected wave will be in phase. In the drawing on the right, you see an erect pulse traveling into the boundary being reflected with no phase change. The pulse went in erect and came out erect. Water waves reflecting off a solid wall are a good example of free end reflection.

Supplemental: This Phet simulation can be extremely helpful for visualizing this and a lot of other topics in these notes.

Reflection at a Non-Rigid BoundaryFrom Barron’s AP Prep

Video Explanation: Twu Waves 6 This is a great video. She is going to walk you through the process you learned above, but go into much greater detail. This will be very helpful not only in this unit, but in the light and sound unit that follows.

Superpositionfrom Barron’s AP Prep

from Knapp Notes

+ =

Const r uct ive int er f er ence

+ =

Dest r uct ive int er f er ence

cancellat ion

There is a typo here, though I like the picture. The one on the right should say “In Phase”. Waves that are in phase have their amplitudes add, waves out of phase have their amplitudes subtract.

Before moving on, make sure you go try a couple superposition problems in the practice section. The next section will be confusing if you don’t make sure you have this down.

Standing WavesSo far, all the types of waves we discussed have been traveling waves, waves appear to be moving along a string or medium. Now we are going to talk about standing waves.

What happens if a wave reflects and ends up interfering with itself? In other words, what if the wave that was sent down bounces off and interacts with the waves heading toward the boundary? Then the wave interferes with itself. The easy way to see this is to open up that Phet simulation I discussed earlier.

Virtual Demonstration: Open up the simulation.

1. Set damping to zero. This means we won’t have any friction or air resistance stealing energy out of the system.2. Play with the program for a moment if you haven’t already. Send a couple of pulses down. Check out the law of

superposition and reflection in action. The more you play with it, the more intuitive these ideas will feel.3. Set the end to be Fixed End.

v v

v

v

M ee tin g w avesou t o f ph ase

M ee tin g w avesin o f ph ase

4. Set it to Oscillate. Set the amplitude to 10 and the frequency to 15. The restart the system. You can watch the returning wave constructively destructively interfere with itself.

5. Now set it the frequency to 25 and restart the system. Here the wave is constantly constructively and destructively interfering with itself. We’ve created a wave that looks like it is standing still. The oscillator on the end is adding more and more energy- sort of like pushing someone on a swing to a particular rhythm. We have created a standing wave.

Video Animation: This great animation from Wikipedia shows a wave reflecting back and interfering with itself to create a standing wave. The red wave is the reflecting wave. The light blue wave is the incident wave. The dark blue wave is the standing wave.

from AP Prep

Video Demonstration and Lecture: Twu Waves 10 Meet Ms. Twu’s son! This is a great video showing the creation of standing waves, and how they are measured. Pause at 3:39. Read the notes below.

Standing Waves and ResonanceResonance occurs when we hit that sweet spot in the frequency where the waves will create a standing wave. If you add energy in at the right frequency, you can counteract energy lost the air resistance and get the wave to sort of sit there permanently. This is how a jump rope works. You hit the right rhythm and you create a have a wavelength of a standing

wave. You actually created the 1st harmonic of the jump rope (see below)

If you increase the frequency (like Ms. Twu’s son did in the video) you will go out of rhythm for awhile and the wave will appear chaotic. Eventually, however, you will hit another resonance frequency and a standing wave with two antinodes and three nodes will be created.

You can keep going up and up in frequency, finding different standing waves.

Harmonic Wavelengths Number of antinodes Number of nodes1 ½ 1 22 1 2 33 1.5 3 44 2 4 5

How do you memorize this stuff? Well take a look- the name of the harmonic tells us the number of antinodes in the standing wave. I think the easiest way to answer questions about this is to just draw it and look at stuff. It’s worked for me for years.

Another, more mathematical way of thinking about is to use this equation.

L=nλ2

Where l is the length of the rope, n is the number of the harmonic, lambda is wavelength.

Calculating Resonant FrequenciesThat very first frequency- the one that created the first harmonic- is called the fundamental frequency (f1).

from Princeton Review

AP Practice: Finding Harmonic Numbers and Wavelengths

Solution

Practice: Nodes and Antinodes

Solution