Chapter 16, Problem 16.1CYU

Check Your Understanding When a guitar string is plucked, the guitar string oscillates as a result of waves moving through the string. The vibrations of the string cause the air molecules to oscillate, forming sound waves. The frequency of the sound waves is equal to the frequency of the vibrating string. Is the wavelength of the sound wave always equal to the wavelength of the waves on the string?

To determine

Whether or not, the wavelength of the sound wave is always equal to the wavelength of the waves on the string.

Answer to Problem 16.1CYU

No, the wavelengths of the sound wave and the wave on the string will be different even if they have the same frequency.

Explanation of Solution

Introduction:

The relation between the speed of the wave $(v)$, frequency $(f)$ and the wavelength $\lambda$ is given by

$v=f\times \lambda$ …………… (1)

The speed of the wave on the string is given beloW

$v=\sqrt{\frac{Tensioninthestring\left(T\right)}{linearmassdensity\left(\mu \right)}}$ …………. (2)

The speed of the sound wave is

$v=331\left(\frac{m}{\sec }\right)\times \sqrt{\frac{Temperatureoftheair\left(Kelvin\right)}{273}}$ …………. (3)

The speed of waves on the string is given by equation (2).It depends on the tension in the string and the linear mass density of the string. On the other hand, speed of the sound waves depends on the temperature of the air.

Even if the frequency of both the waves, on the string and in the air is same, their speeds would be different.

According equation (1), for the same frequency and different speed, wavelength would be different.

Hence, the speed of the both the waves are different which results in different wavelength even if the frequency is same.

Note: To obtain the same wavelength, one need to adjust the tension in the string and the linear mass density of the string to the temperature of the air so that the speed of both the waves, on the string and in the air, is same.

Conclusion:

No, the wavelength would be different of both the waves, on the string and sound wave in air due different speed.