A transverse wave propagating along x-axis is represented by: `y(x,t)=8.0sin(0.5pix-4pit-(pi)/(4))` Where `x` is in metres and `t` is in seconds. The speed of the wave is:

Options

A. `4pi m//s`
B. `0.5pi m//s`
C. `pi/4 m//s`
D. 8 m/s

Answer

To find the speed of the transverse wave represented by the equation \( y(x,t) = 8.0 \sin(0.5\pi x - 4\pi t - \frac{\pi}{4}) \), we can follow these steps:

### Step 1: Identify the wave parameters
The standard form of a wave equation is given by:
\[ y(x,t) = A \sin(kx - \omega t + \phi) \]
From the given equation, we can identify:
- Amplitude \( A = 8.0 \)
- Wave number \( k = 0.5\pi \)
- Angular frequency \( \omega = 4\pi \)
- Phase constant \( \phi = -\frac{\pi}{4} \)

### Step 2: Calculate the wavelength \( \lambda \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{2\pi}{k} \]
Substituting \( k = 0.5\pi \):
\[ \lambda = \frac{2\pi}{0.5\pi} = \frac{2}{0.5} = 4 \text{ meters} \]

### Step 3: Calculate the period \( T \)
The angular frequency \( \omega \) is related to the period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Rearranging this gives:
\[ T = \frac{2\pi}{\omega} \]
Substituting \( \omega = 4\pi \):
\[ T = \frac{2\pi}{4\pi} = \frac{1}{2} \text{ seconds} \]

### Step 4: Calculate the wave speed \( v \)
The speed \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\lambda}{T} \]
Substituting the values of \( \lambda \) and \( T \):
\[ v = \frac{4 \text{ m}}{0.5 \text{ s}} = 8 \text{ m/s} \]

### Final Answer
The speed of the wave is \( 8 \text{ m/s} \).

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