Question:
The displacement of an elastic wave is given by the function y = 3 sin ωt + 4 cos ωt where y is in cm and t is in second. Calculate the resultant amplitude.
Solution:
Given,
y = 3 sin ωt + 4 cos ωt
Let’s consider
3 = a cos φ
4 = a sin φ
Then
y = a cos φ ωt + a sin φ ωt
y = a sin (ωt + φ)
tan φ = 4/3
φ = tan-1 4/3
When the above equation is squared, we het
a2 = 25
a = 5
Therefore, the new amplitude is 5 cm.