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# Waves Study Guide (page 2)

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Updated on Sep 27, 2011

#### Example 1

A microwave oven works based on microwaves that have a wavelength between 30 cm and 10–3 m. Find the range of frequencies if the speed is 3 · 108 m/s.

#### Solution 1

We need the frequencies, so the previous definition of speed will be needed, and then we can solve for frequency in two cases:

λl 30cm = 3 · 10–l m

λ2 10–3 m

v = 3 · 108 m/s

f1 = ?

f2 = ?

v = f · λ

f =

f1 = = = 1 · 109 Hz

and

f2 = = = 3 · 1011 Hz

So, the range of frequencies is between 109 and 3 · 1011Hz.

Mechanical waves in the examples before need a medium in which to propagate: for example, water, string, a spring, and so on. The speed of propagation though is different according to the intrinsic properties of the material. The same interaction, or force, can produce different frequency waves depending on the characteristics of the string. An expression of the speed in a string shows speed to be:

v =

In this expression, T is the tension created in the material through interaction, and m/ L is called linear density and refers to the mass per unit of length.

#### Example 2

Consider a string acted upon by a tension of 152 N and having a length of 52 cm and a mass of 134 g. Find the speed of the wave propagating through the cord.

#### Solution 2

First, look for known data and SI units. Then, set your equation for wave propagation and solve for the unknown.

T = 152 N

L = 52 m = 0.52 m

m = 134 g = 0.134 kg

v = ?

v = =

v = √590 s2/m2

v = 24 m/s

## Sound Waves

One special type of wave that can be created mechanically and propagates through matter is called sound. Although the frequency range for sound is rather large, human beings can hear sounds only between about 12 and 20 Hz and between 14 and 20 KHz. Below the 20 Hz extreme, frequencies are called infrasonic, whereas above 20 KHz, they are called ultrasonic.

We hear sounds not only with different frequencies but also with different intensities. The loudness of the sound is given (measured) by the amplitude of the wave. And the amplitude of the wave is given (measured) by the pressure change with the wave.

As mentioned previously, in order for sound to propagate, a medium is needed—be it a gas, a liquid, or a solid. Sound will not propagate in a vacuum. The reason for this is that sound waves propagate by creating regions of compression (or condensation) and rarefaction in the medium where the sounds acts. Once propagation has started, the change of pressure due to the sound waves has a pattern similar to the other waves we studied previously (see Figure 17.7).

As with the case of waves in strings, the speed of sound is affected by the intrinsic properties of the material. We will define three different speed expressions— one for gas, one for liquids, and one for solids.

When a disturbance is produced, particles collide with each other forming a region of high concentration in the direction of propagation (condensation or compression area) and leaving behind a region of low density (rarefaction). Hence, the disturbance and the wave are in the same direction, and we conclude that sound waves are longitudinal waves.

Experimental work shows that the speed of sound in gases (ideal gases) is:

v =

In this expression, the γ is the adiabatic factor and depends on the material, T is the temperature in Kelvin, m is the mass of a molecule of substance in the specific medium, and k is Boltzmann's constant and is equal to 1.38 · 10–23 J/K.

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