A student standing in a canyon yells “echo” and her voice produces a sound wave of the frequency of f=0.54 kHz. The echo takes t=4.8 s to return to the student. Assume the speed of sound through the atmosphere at this location is v=328 m/s

A student standing in a canyon yells “echo” and her voice produces a sound wave of the frequency of f=0.54 kHz. The echo takes t=4.8 s to return to the student. Assume the speed of sound through the atmosphere at this location is v=328 m/s

What is the wavelength of the soundwave in meters?

Input the expression for the distance, $d$, the canyon wall is from the student. Answer should look like d=.

This question aims to find the wavelength of the soundwave and the expression for the distance traveled by the sound.

Sound is a mechanical wave produced by the back-and-forth vibration of the particles in the medium by which the sound wave travels. It is a vibration that travels as an acoustic wave through a medium such as solid, liquid, or gas.

The vibration of an object results in the vibration of the air molecules as well, causing a chain reaction of sound wave vibrations to travel throughout the medium. This constant back-and-forth motion creates a low and high-pressure region in the medium. Compressions refer to the high-pressure and rarefactions refer to the low-pressure regions, respectively. The number of compressions and rarefactions that take place per unit of time is said to be the frequency of soundwave.

Expert Answer

Â Here is the expert answers for this question along with clear explanations.

For Wavelength:

The variation of pressure in a sound wave continues to repeat itself over a specific distance. This distance is referred to as the wavelength. In other words, the wavelength of a sound is the distance between successive compression and rarefaction and the period is the time it takes to complete one cycle of the wave.

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Given data is:

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$f=0.45\,kHz$Â orÂ $540\, Hz$

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$t=4.8\,s$

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$v=328\,m/s$

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Here, $f,t$Â and $v$ refer to frequency, time, and velocity, respectively.

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Let $\lambda$ be the wavelength of the soundwave, then:

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$\lambda=\dfrac{v}{f}$

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$\lambda=\dfrac{328\,m/s}{540\,Hz}=0.61\,m$

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For Distance:

Let $d$ be the distance of the canyon wall from the student, then:

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$d=\dfrac{vt}{2}$

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$d=\dfrac{382\times 4.8}{2}=787.2\,m$

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Example 1

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Find the speed of sound when its wavelength and frequency are measured as:

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$\lambda=4.3\,m$Â andÂ $t=0.2\,s$.

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Since,Â $f=\dfrac{1}{t}$

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$f=\dfrac{1}{0.2\,s}=5\,s^{-1}$

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Also, as:

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$\lambda=\dfrac{v}{f}$

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$\implies v=\lambda f $

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So, $v=(4.3\,m)(5\,s^{-1})=21.5\,m/s$

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Example 2

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A wave travels at $500\, m/s$ in a specific medium. Calculate the wavelength if $6000$ waves pass over a specific point of the medium in $4$ minutes.

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Let $v$ be the speed of the wave in the medium, then:

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$v=500\,ms^{-1}$

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Frequency $(f)$ ofÂ wave $=$ Number of waves passing per second