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# Longitudinal Wave

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Longitudinal Waves
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Playing with a Slinky is a childhood tradition, but few children realize they are actually playing with physics.

### Longitudinal Waves

Like transverse waves, longitudinal waves  are mechanical waves, which means they transfer energy through a medium. Unlike transverse waves, longitudinal waves cause the particles of medium to move parallel to the direction of the wave. They are most common in springs, where they are caused by the pushing an pulling of the spring. Although the surface waves on water are transverse waves, fluids (liquids, gases, and plasmas) usually transmit longitudinal waves.

As shown in the image below, longitudinal waves are a series of compressions and rarefactions , or expansions. The wavelength of longitudinal waves is measured by the distance separating the densest compressions. The amplitude of longitudinal waves is the difference in media density between the undisturbed density to the highest density in a compression.

Example Problem: A sonar signal (sonar is sound waves traveling through water) of  $1.00 \times 10^6$ Hz frequency has a wavelength of 1.50 mm in water.  What is the speed of sound in water?

Solution: $v=\lambda f =(0.00150 \ \text{m})(1.00 \times 10^6 \ \text{s}^{-1})=1500 \ \text{m/s}$

Example Problem: A sound wave of wavelength 0.70 m and velocity 330 m/s is produced for 0.50 s.

1. What is the frequency of the wave?
2. How many complete waves are emitted in this time interval?
3. After 0.50 s, how far is the wave front from the source of the sound?

Solution:

1. $f=\frac{v}{\lambda}=\frac{330 \ \text{m/s}}{0.70 \ \text{m}}=470 \ \text{s}^{-1}$
2. $\text{complete waves} = (470 \ \text{cycles/s})(0.50 \ \text{s}) = 235 \ \text{cycles}$
3. $\text{distance} = (330 \ \text{m/s})(0.50 \ \text{s}) = 115 \ \text{m}$

#### Summary

• Longitudinal waves cause the particles of medium to move parallel to the direction of the wave.

#### Practice

The following video explains how a tuning fork creates sound . Use this resource to answer the questions that follow.

1. In your own words, how are compressions and rarefactions produced by the tuning fork?
2. Make a guess why sound can easily travel around corners (Hint: think of its medium).

#### Review

1. Some giant ocean waves have a wavelength of 25 m long, and travel at speeds of 6.5 m/s. Determine the frequency and period of such a wave.
2. Bats use sound echoes to navigate and hunt.  They emit pulses of high frequency sound waves which reflect off obstacles in the surroundings.  By detecting the time delay between the emission and return of a pulse, a bat can determine the location of the object.  What is the time delay between the sending and return of a pulse from an object located 12.5 m away?  The approximate speed of sound is 340 m/s.
3. Sachi is listening to her favorite radio station which broadcasts radio signals with a frequency of $1.023 \times 10^8 \ \text{Hz}$ .  If the speed of the signals in air is $2.997 \times 10^8 \ \text{m/s}$ , what is the wavelength of these radio signals?
4. A longitudinal wave is observed to be moving along a slinky.  Adjacent crests are 2.4 m apart.  Exactly 6 crests are observed to move past a given point in 9.1 s.  Determine the wavelength, frequency, and speed of this wave.
5. A sonar signal leaves a submarine, travels through the water to another submarine and reflects back to the original submarine in 4.00 s.  If the frequency of the signal was 512 cycles per second and the wavelength of the signal was 2.93 m, how far away is the second submarine?