For the wave described in Exercise 14.8, plot the displacement versus graphs for and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?
(difference) Phase differs from one point to another in a travelling wave, while amplitude and frequency remain the same.
(graphs) For x = 0: y = 3.0 sin(36t + π/4) For x = 2 cm: y = 3.0 sin(36t + 0.036 + π/4) For x = 4 cm: y = 3.0 sin(36t + 0.072 + π/4)
(shapes) All three graphs are sinusoidal curves (sine waves) with the same amplitude of 3.0 cm and same frequency.
Explanation
When we fix a position x in the wave equation y(x,t) = 3.0 sin(36t + 0.018x + π/4), the equation becomes a function of time only, representing simple harmonic motion at that point. Substituting x = 0, 2, and 4 cm gives three sinusoidal equations with identical amplitude (3.0 cm) and angular frequency (ω = 36 rad/s), but different phase constants. The phase at each point depends on the term 0.018x, so points at different positions have different phases. As stated in the context, 'the particle at the origin oscillates about its mean position as the wave progresses' and 'this is true for any other location also' - meaning all points oscillate with same amplitude and frequency, only phase differs.
Solution Steps
Step 1: For a fixed position x, the wave equation y(x,t) = 3.0 sin(36t + 0.018x + π/4) becomes a function of time only.
Step 2: Substitute x = 0: y = 3.0 sin(36t + π/4)
Step 3: Substitute x = 2 cm: y = 3.0 sin(36t + 0.036 + π/4)
Step 4: Substitute x = 4 cm: y = 3.0 sin(36t + 0.072 + π/4)
Step 5: All three graphs are sinusoidal with same amplitude (3.0 cm) and frequency, but different phases.
Step 6: In travelling waves, particles at different positions oscillate with same amplitude and frequency, but different phases.