Beat and Displacement Frequency

## Question:

It is said in the textbook that when two waves of slightly different frequencies f1 & f2 (but of equal amplitudes) are travelling in the same direction and of the same speed, there will be superposition. We can measure its beat frequency and displacement frequency to determine whether it is a loud voice or a low voice.

Q1: What are 'beat frequency' & 'displacement frequency'?'

Q2: Why the beat frequency = f1 - f2, but the displacement frequency = f1 + f2 / 2? Why we divided it latter one by two?

## Answer:

It is hard for me to know how much detail to include in answering your questions. You are getting into an area where a knowledge of trigonometry will be a great help. I will try to answer in the most simple way. If you need more detail please let me know.

When two waves of nearly the same frequency and amplitude are traveling in the same direction, an observer at a fixed point in space would detect a wave that is the superposition of the two original waves. Since the two waves differ slightly in frequency, the superposition amplitude at a fixed point will vary with the passage of time. When the waves are nearly in phase. the amplitude will be large. When the waves are nearly out of phase the amplitude will be small.

The displacement frequency is the apparent frequency of the superposition wave. That will be the average frequency of the two component waves or (f1+f2)/2. The amplitude of the superposition wave will be varying in time in accordance with the expression A=2*A0*cos(2*p*((f1-f2)/2)*t), where A0 is the amplitude of each of the component waves. Getting to that expression is where the trigonometry comes in.

A person listening to the superposition wave would hear a maximum in amplitude at the point where cos(2*p*((f1-f2)/2)*t) equals +1 and another where cos(2*p*((f1-f2)/2)*t) equals -1. That results in two beats per cycle so the beat frequency, which is the frequency of the changes in amplitude, is two times (f1-f2)/2 or just f1-f2.