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# Wave Equation

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Students will learn how to use the wave equation to solve problems.

### Key Equations

$T = \frac{1}{f} \; \; \text{Wave period}$

$v = \lambda f \; \; \text{Wave velocity}$

### Watch this Explanation

Guidance
The wave equation is analogous to distance equals rate multiplied by time equation for one cycle. Here the distance is a wavelength and the time is the period. We simply divide both sides by the period, to get speed of the wave equals the wavelength multiplied by one over the period. This is equivalent to wavelength multiplied by frequency, since frequency equals one of the period by definition.

#### Example 1

While on vacation in Hawaii you observe waves at the Banzai Pipeline approaching the shore at 6.0 m/s. You also note that the distance between waves is 28 m. Calculate (a) the frequency of the waves and (b) the period.

Question a : $f= ? [Hz]$

Given: $v = 6.0 \ m/s$

${\;} \qquad \quad \lambda = 28 \ m$

Equation: $v = f \cdot \lambda$ therefore $f = \frac{v}{\lambda}$

Plug n’ Chug: $f = \frac{v} {\lambda} = \frac{6.0 \ m/s}{28 \ m} = 0.21 \ Hz$

Answer: $\boxed{\mathbf{0.21 \ Hz}}$

Question b : $T = ? [s]$

Given: $f = 0.21 \ Hz$

Equation: $T = \frac{1}{f}$

Plug n’ Chug: $T = \frac{1}{f} = \frac{1}{0.21 \ Hz} = 4.76 \ s$

Answer: $\boxed{\mathbf{4.76 \ s}}$

### Explore More

1. Bored in class, you start tapping your finger on the table. Your friend, sitting right next to you also starts tapping away. But while you are tapping once every second, you’re friend taps twice for every one tap of yours.
1. What is the Period and frequency of your tapping?
2. What is the Period and frequency of your friend’s tapping?
3. Your tapping starts small waves going down the desk. Sort of like hitting a bell with a hammer. The frequency of the sound you hear is 1200 Hz. You know the wave speed in wood is about 3600 m/s. Find the wavelengths generated by your tapping.
2. You’re sitting on Ocean Beach in San Francisco one fine afternoon and you notice that the waves are crashing on the beach about 6 times every minute.
1. Calculate the frequency and period of the waves.
2. You estimate that it takes 1 wave about 4 seconds to travel from a surfer 30 m off shore to the beach. Calculate the velocity and average wavelengths of the wave.
3. The Sun tends to have dark, Earth-sized spots on its surface due to kinks in its magnetic field. The number of visible spots varies over the course of years. Use the graph of the sunspot cycle below to answer the following questions. (Note that this is real data from our sun, so it doesn’t look like a perfect sine wave. What you need to do is estimate the best sine wave that fits this data.)
1. Estimate the period $T$ in years.
2. When do we expect the next “solar maximum?”
4. Human beings can hear sound waves in the frequency range 20 Hz – 20 kHz. Assuming a speed of sound of 343 m/s, answer the following questions.
1. What is the shortest wavelength the human ear can hear?
2. What is the longest wavelength the human ear can hear?
5. The speed of sound in hydrogen gas at room temperature is $1270\;\mathrm{m/s}$ . Your flute plays notes of $600, 750,$ and $800\;\mathrm{Hz}$ when played in a room filled with normal air. What notes would the flute play in a room filled with hydrogen gas?
6. The speed of light $c$ is 300,000 km/sec.
1. What is the frequency in Hz of a wave of red light $(\lambda = 0.7 \times 10^{-6} \ m)$ ?
2. What is the period $T$ of oscillation (in seconds) of an electron that is bouncing up and down in response to the passage of a packet of red light? Is the electron moving rapidly or slowly?
7. Radio signals are carried by electromagnetic waves (i.e. light waves). The radio waves from San Francisco radio station KMEL (106.1 FM) have a frequency of 106.1 MHz. When these waves reach your antenna, your radio converts the motions of the electrons in the antenna back into sound.
1. What is the wavelength of the signal from KMEL?
2. What is the wavelength of a signal from KPOO (89.5 FM)?
3. If your antenna were broken off so that it was only 2 cm long, how would this affect your reception?

1. a. 1.0s, 1 Hz b. 0.5s, 2 Hz c. 3 m
2. a. 0.1 Hz, 10.0 s b. 7.5 m/s, 75 m

1. a. 1.7 cm b.17 m
2. $2230 \;\mathrm{Hz}; 2780 \;\mathrm{Hz}; 2970 \;\mathrm{Hz}$
3. $4.3 \times 10^{14} \ Hz$
1. $2.3 \times 10^{-15} \ s$
4. a. 2.83 m b. 3.35 m c. rule of thumb, antenna should be $\frac{1}{4} \lambda$ , thus quality of reception will suffer