Question

The equation of a stationary wave on a string fixed at both ends is given by




y( x,t) = 2sin πx cos100πt




where x and y are measured in metre and t in second. Calculate the amplitude,




wavelength and frequency of component waves whose superposition generated this




stationary wave. Also write the equations of component waves.

Answer 1

y(x, t) = 2 sin(πx) cos(100πt) = sin(πx+100πt)+sin(πx−100πt) = sin(πx+100πt)+sin(100πt−πx+π)

General formula for a standing wave: y(x, t) = A sin(ωt − kx + φ).

We have y(x, t) = y1(x, t) + y2(x, t).

Consider first component y1(x, t) = sin(πx + 100πt):

amplitude is A1 = 1 m

angular frequency is ω1 = 100π rad/s

frequency is f1 = ω1/2π = 50 Hz

wave number is k1 = −π rad/m

velocity is v1 = ω1/k1 = −100 m/s

(velocity is negative because the wave is travelling in the negative x direction)

wavelength is λ1 = |v1|/f1 = 2 m

Consider second component y2(x, t) = sin(100πt − πx + π):

amplitude is A2 = 1 m

angular frequency is ω2 = 100π rad/s

frequency is f2 = ω2/2π = 50 Hz

wave number is k2 = π rad/m

velocity is v2 = ω2/k2 = 100 m/s

wavelength is λ2 = |v2|/f2 = 2 m