Question

Given below are some functions of x and t to represent the displacement of an elastic wave.

1. y = 5 cos (4x) sin (20t)
2. y = 4 sin (5x – t/2) + 3 cos (5x – t/2)
3. y = 10 cos [(252 – 250) πt] cos [(252 + 250)πt]
4. y = 100 cos (100πt + 0.5x)

State which of these represent

1. a travelling wave along –x direction
2. a stationary wave
3. beats
4. a travelling wave along +x direction.

Given reasons for your answers.

Answer 1

1. The equation y = 100 cos(100πt + 0.5x) is representing a travelling wave along the x-direction.
2. The equation y = 5 cos(4x) sin(20t) represents a stationary wave because it contains sin, cos terms i.e., the combination of two progressive waves
3. As the equation y = 10 cos[(252 – 250)πt] – cos[(252 + 250)πt] involving the sum and difference of two nearby frequencies 252 and 250 this equation represents beats formation.
4. As the equation, y = 4 sin(5x – t/2) + 3 cos(5x – t/2) involves a negative sign with x, have if represents a travelling wave along the x-direction.