To play a $\mathrm{C}$ major chord on the piano, you hit the $\mathrm{C}, \mathrm{E}$, and $\mathrm{G}$ keys simultaneously. When you do so, you hear no beats. Why?

When a $63 \mathrm{~cm}$-long guitar string is plucked, all possible standing waves simultaneously exist on the string. When you place your finger on the string, any standing waves with nonzero amplitude at that location are stopped, but any standing waves with a node at that location continue vibrating. If you wish to stop all standing waves with frequencies lower than the third harmonic on this string, how far from the end of the string should you place your finger?

When a $63 \mathrm{~cm}$ long guitar string is pluciked, all possible standing waves simultaneously exist on the string. When you place your finger on the string, any standing waves with nonzero amplitude at that location are stopped, but any standing waves with a node at that location continue vibrating. If you wish to stop all standing waves with frequencies lower than the third harmonic on this string, how far from the end of the string should you place your finger?

Four waves are described by the following equations, in which all distances are measured in centimeters and all times are measured in seconds:

$$\begin{aligned} &y_{\mathrm{A}}=10 \cos (3 x-4 t) \\ &y_{\mathrm{B}}=10 \cos (5 x+4 t) \\ &y_{\mathrm{C}}=20 \cos (-10 x+60 t) \\ &y_{\mathrm{D}}=20 \cos (-4 x-20 t) \end{aligned}$$

Which wave has the greatest speed?

Four waves are described by the following equations, in which all distances are measured in centimeters and all times are measured in seconds:

$$\begin{aligned} &y_{\mathrm{A}}=10 \cos (3 x-4 t) \\ &y_{\mathrm{B}}=10 \cos (5 x+4 t) \\ &y_{\mathrm{C}}=20 \cos (-10 x+60 t) \\ &y_{\mathrm{D}}=20 \cos (-4 x-20 t) \end{aligned}$$

Which wave has the greatest wavelength?

Four waves are described by the following equations, in which all distances are measured in centimeters and all times are measured in seconds:

$$\begin{aligned} &y_{\mathrm{A}}=10 \cos (3 x-4 t) \\ &y_{\mathrm{B}}=10 \cos (5 x+4 t) \\ &y_{\mathrm{C}}=20 \cos (-10 x+60 t) \\ &y_{\mathrm{D}}=20 \cos (-4 x-20 t) \end{aligned}$$

Which wave has the highest frequency?

Four waves are described by the following equations, in which all distances are measured in centimeters and all times are measured in seconds:

$$\begin{aligned} &y_{\mathrm{A}}=10 \cos (3 x-4 t) \\ &y_{\mathrm{B}}=10 \cos (5 x+4 t) \\ &y_{\mathrm{C}}=20 \cos (-10 x+60 t) \\ &y_{\mathrm{D}}=20 \cos (-4 x-20 t) \end{aligned}$$

Which of these waves travel in the $-x$ direction?

On a rainy day, while driving your car, you notice that your windshield wipers are moving in synchrony with the wiper blades of the car in front of you. After several cycles, however your wipers and the wipers of the other car are moving opposite to one another. A short time later the wipers are synchronous again. What wave phenomena do the wipers illustrate? Explain.

Four waves are described by the following equations, in which all distances are measured in centimeters and all times are measured in seconds:

$$\begin{aligned} &y_{\mathrm{A}}=10 \cos (3 x-4 t) \\ &y_{\mathrm{B}}=10 \cos (5 x+4 t) \\ &y_{\mathrm{C}}=20 \cos (-10 x+60 t) \\ &y_{\mathrm{D}}=20 \cos (-4 x-20 t) \end{aligned}$$

Which of these waves travel in the $+x$ direction?

Identical cellos are being tested. One is producing a fundamental frequency of $120.9 \mathrm{~Hz}$ on a string that is $1.25 \mathrm{~m}$ long and has a mass of $109 \mathrm{~g}$. On the second cello an identical string is fingered to reduce the length that can vibrate. If the beat frequency produced by these two strings is $5.25 \mathrm{~Hz}$, what is the vibrating length of the second string?

A surfer floating beyond the breakers notes 14 waves per minute passing her position. If the wavelength of these waves is $34 \mathrm{~m}$, what is their speed?

A wave on a string is described by the following equation:

$$y=(15 \mathrm{~cm}) \cos \left(\frac{\pi}{5.0 \mathrm{~cm}} x-\frac{\pi}{12 \mathrm{~s}} t\right)$$

In which direction does the wave travel?

Two musicians are comparing their clarinets. The first clarinet produces a tone that is known to be $466 \mathrm{~Hz}$. When the two clarinets play together they produce eight beats every $3.00$ seconds. If the second clarinet produces a higher pitched tone than the first clarinet, what is the second clarinet's frequency?

A wave on a string is described by the following equation:

$$y=(15 \mathrm{~cm}) \cos \left(\frac{\pi}{5.0 \mathrm{~cm}} x-\frac{\pi}{12 \mathrm{~s}} t\right)$$

What is the speed of the wave?