Question:

A wave with a frequency of 500 Hz is travelling through a medium. The wave has a wavelength of 0.4 m and an amplitude of 0.2 m. Determine the following characteristics of the wave:

a) Wave speed b) Period c) Angular frequency d) Phase constant e) Total energy carried by the wave

Answer:

a) Wave speed can be calculated using the formula:

Wave Speed = Frequency × Wavelength

Given: Frequency (f) = 500 Hz Wavelength (λ) = 0.4 m

Substituting the values into the formula, we get:

Wave Speed = 500 Hz × 0.4 m
Wave Speed = 200 m/s

Therefore, the wave speed is 200 m/s.

b) Period can be calculated using the formula:

Period = 1 / Frequency

Given: Frequency (f) = 500 Hz

Substituting the value into the formula, we get:

Period = 1 / 500 Hz
Period = 0.002 s

Therefore, the period is 0.002 s.

c) Angular frequency (ω) can be calculated using the formula:

Angular Frequency = 2π × Frequency

Given: Frequency (f) = 500 Hz

Substituting the value into the formula, we get:

Angular Frequency = 2π × 500 Hz
Angular Frequency = 3141.59 rad/s (rounded to two decimal places)

Therefore, the angular frequency is 3141.59 rad/s.

d) Phase constant (φ) is not given in the question, so we cannot determine it without additional information or context.

e) The total energy carried by the wave can be calculated using the formula:

Total Energy = (1/2) × (Wave Speed^2) × (Waveform Area Density)

The waveform area density is defined as the product of the amplitude squared and the mass per unit length of the medium. However, since the mass per unit length is not provided in the question, we cannot calculate the total energy without further information.

Therefore, the total energy carried by the wave cannot be determined with the given information.

Note: The phase constant (φ) and total energy depend on additional information that is not provided in the given question. Therefore, we cannot calculate them accurately based on the given data alone.