Question

A periodic wave is composed of the superposition of three sine waves whose frequencies are \(36,60,\) and 84 Hz. The speed of the wave is 180 m/s. What is the wavelength of the wave? [Hint: The \(36 \mathrm{Hz}\) is not necessarily the fundamental frequency.]

Short answer

Answer: The wavelength of the periodic wave is 15 meters.

Step-by-step solution

Find the Fundamental Frequency

Note that the three given frequencies are 36, 60, and 84 Hz. We want to find the fundamental frequency, which is the lowest frequency that divides all of these frequencies. Let's find the greatest common divisor (GCD) of these three numbers, as it corresponds to the fundamental frequency: GCD(36, 60, 84) = 12 Hz So, the fundamental frequency is 12 Hz.

Calculate the Wavelength

We can now use the wave speed formula to calculate the wavelength of the wave. The wave speed formula is given by: speed = frequency × wavelength We know the wave speed is 180 m/s, and we just found the fundamental frequency to be 12 Hz. We can rearrange the formula to find the wavelength: wavelength = speed / frequency Now, we can plug in the values: wavelength = 180 m/s / 12 Hz Wavelength = 15 meters The wavelength of the wave is 15 meters.