Question 14 of 23intermediate🔧 ApplyNumerical4 marks

A transverse harmonic wave on a string is described by y(x,t)=3.0sin⁡(36t+0.018x+π/4)y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4) where xx and yy are in cm and tt in s. The positive direction of xx is from left to right.

(a)

Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation?

Answer

This is a travelling wave. Speed = 2000 cm/s (or 20 m/s). Direction: From right to left (negative x-direction).

(b)

What are its amplitude and frequency?

Answer

Amplitude = 3.0 cm. Frequency = 18/π Hz ≈\approx 5.73 Hz.

(c)

What is the initial phase at the origin?

Answer

Initial phase at the origin = π/4 radians.

(d)

What is the least distance between two successive crests in the wave?

Answer

Least distance between two successive crests = 1000π/9 cm ≈\approx 349 cm ≈\approx 3.49 m.

Explanation

The wave equation is compared with the standard form y = a sin(ωt + kx + φ). For part (a), the presence of (kx + ωt) combination indicates a travelling wave moving in negative x-direction. For part (b), amplitude and angular frequency are directly read from the equation. For part (c), substituting x = 0 gives the phase at origin. For part (d), wavelength is calculated using λ = 2π/k.

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