Exam Question

A wave is described by the equation y(x,t) = 0.05 sin(2π(4t - 3x/5)). Determine the wave characteristics based on this equation.

Solution

We are given the equation of a wave as y(x,t) = 0.05 sin(2π(4t - 3x/5)).

To determine the wave characteristics, we need to identify the amplitude, wavelength, frequency, and speed of the wave.

Step 1: Find the amplitude (A) of the wave:

The amplitude (A) of a wave is the maximum displacement of any point on the wave from its equilibrium position.

In this equation, the amplitude is given as 0.05.

Therefore, the amplitude of the wave is 0.05.

Step 2: Find the wavelength (λ) of the wave:

The wavelength (λ) of a wave is the distance between two consecutive points on the wave that are in phase.

The equation for wavelength is given by:

λ = (speed of the wave) / (frequency of the wave)

In our equation, the frequency is given as 4.

The wave equation is in the form y(x,t) = A sin(2πft).

Comparing with our equation, we can determine the frequency (f) as 4.

Therefore, the wavelength (λ) of the wave can be found by substituting the frequency into the equation as:

λ = (speed of the wave) / (4)

Since the speed of the wave is not given, we cannot determine the exact wavelength without additional information. However, we can conclude that the wavelength is inversely proportional to the frequency, meaning that as the frequency increases, the wavelength decreases, and vice versa.

Step 3: Find the frequency (f) of the wave:

The frequency (f) of a wave is the number of complete oscillations or cycles the wave goes through in one second.

In our equation, the frequency is given as 4.

Therefore, the frequency of the wave is 4 Hz.

Step 4: Find the speed (v) of the wave:

The speed (v) of a wave is the distance traveled by one complete cycle of the wave in one unit of time.

In our equation, the speed of the wave is not explicitly given. However, we can relate the speed (v), frequency (f), and wavelength (λ) of a wave using the equation:

v = fλ

Substituting the given values:

v = 4 * λ

Since we do not have the exact value of the wavelength, we cannot determine the speed of the wave without additional information.

In conclusion:

The amplitude of the wave is 0.05. The frequency of the wave is 4 Hz. The wavelength and speed of the wave cannot be determined without additional information.

Answer: The amplitude of the wave is 0.05. The frequency of the wave is 4 Hz. The wavelength and speed of the wave cannot be determined without additional information.