How do you calculate the freq of a sine wave?

Answer 1

Suppose a sine wave of the form y = A*sin(k) with

• A = amplitude or maximum value of the function y (namely when k = pi/2 or 90°)
• k = the value on the x-axis of the function

It's typical of a sine wave that it's periodic, which means the function y repeats itself after a certain period. This period is equal to 2*pi or 360°, for example:

for k = pi/2, 5*pi/2, 9*pi/2, ... the value of y will be the same and equal to A (notice that 5*pi/2 = pi/2 + 2*pi and 9*pi/2 = 5*pi/2 + 2*pi)

In physics it's a more common practice to write a sine wave as y = A*sin(omega*t) with omega the angular frequency specified in radians/s (omega refers to the Greek letter) and t the time specified in seconds.

Now, when you want to calculate the frequency f of a sine wave (which is not equal to the angular frequency) or in other words the number of complete cycles that occur per second (specified in cycle/s or s-1 or Hz), you need to know the time T required to complete one full cycle (specified in s/cycle or just s or Hz-1). The frequency f is then equal to 1/T.

Knowing omega you can calculate the frequency in a different and more common way:

since the sine wave is periodic and after a time T one cycle has been completed (thus one period), it follows that omega*T = 2*pi for the function y to have the same value after one period (the function y having the same value is equal to completing one cycle).

Let's rearrange this formula by bringing 2*pi to the left and T to the right, so we get:

omega/(2*pi) = 1/T and since 1/T = f we finally get:

f = omega / (2*pi)