Question

If sine waves of frequencies \[50,100,150,200\] and \[250Hz\] are added together, the resulting wave will be
A) periodic wave
B) a non-periodic wave
C) a chaotic wave
D) an increasing ramp wave

Answer 1

Hint: The addition of sine waves and other sinusoidal signals is done using Fourier series. The very idea of Fourier series is the representation of functions using sine and cosine waves and also to make other functions by adding infinite sine and cosine waves.

Complete step by Step Solution:
The very basic periodic function is represented by a square wave. Let’s consider a sine wave of a certain frequency and a square wave of the same frequency, then the sum of the square wave, the average value of the square wave and the sine wave of the same frequency, all taken together, form a very rough approximation of the actual square wave itself.
As we add sinusoidal waves of increasingly, higher frequency, the addition of the higher frequencies better approximates the rapid changes, or details of the original function (in this case, the square wave). The addition of higher frequency harmonics also approximates the changes in slope in the original function.
From the argument stated above, we can say that, as higher frequency sine waves are added together, the resulting waves resemble an equivalent square wave more and more closely. Therefore we can conclude that the wave resulting from the addition of sine waves of frequencies \[50,100,150,200\] and \[250Hz\] will be a periodic wave.

Hence, the correct option is (A).

Note: Another method to add sine waves is to find the averages of the wavenumbers and the angular frequencies of the waves to be added and then express the resulting wave as a product of sine and cosine functions having the average values as arguments. But that’s applicable if wavenumber and angular frequency are known to us.