Question

# Two travelling sinusoidal waves are described by the wave functions  y1=10sin[π(8x−1400t)] and y2=10sin[π(8x−1400t−0.5)] Where x, y1, and y2 are in metres and t is in seconds . which of the following options is/are correct ?

A
Amplitude of the resultant wave is 102 m
B
Frequency of the resultant wave  is  700 Hz
C
Time period of the resultant wave is 0.1 s
D
None of these

Solution

## The correct options are A Amplitude of the resultant wave is 10√2 m B Frequency of the resultant wave  is  700 HzGiven wave functions are y1=10sin[π(8x−1400 t)]    .....(1) y2=10sin[π(8x−1400t−0.5)]    ......(2) Comparing these with the general equation,  y=Asin(kx−ωt+ϕ)  Phase difference between (1) and (2) is given by ϕ=π2 Now, According to the principle of superposition, the resultant wave function will be : →y=→y1+→y2 From (1) and (2), we can conclude that the waves are moving in the same direction, so we can write the above equation as  yr=y1+y2 ⇒yr=2A0cos(ϕ2)sin(kx−ωt−ϕ2) Using the given data, yr=(2×10)cos(π4)sin[π(8x−1400t)−π4] ⇒yr=10√2sin[π(8x−1400t)−π4] So, Resultant amplitude Ar=10√2 m Resultant Angular frequency ωr=1400π frequency of resultant wave  f=1400π2π=700 Hz ∴ Time period (T)=1f=1700=0.0014 sec  Thus, options (a) and (b) are the correct answers.

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