A wave has an amplitude of 0.02 m, frequency 660 Hz and velocity 330 m/s. If the wave propagates along the negative X-direction what is equation for the wave?

Answer

To find the equation of the wave propagating along the negative x-direction, we will follow these steps:

### Step 1: Identify the given parameters
- Amplitude (A) = 0.02 m
- Frequency (f) = 660 Hz
- Velocity (v) = 330 m/s

### Step 2: Determine the angular frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 660 \text{ Hz} \]
Calculating this gives:
\[ \omega = 1320\pi \text{ rad/s} \]

### Step 3: Determine the wave number (k)
The wave number (k) can be calculated using the formula:
\[ k = \frac{\omega}{v} \]
Substituting the values of ω and v:
\[ k = \frac{1320\pi}{330} \]
Calculating this gives:
\[ k = 4\pi \text{ rad/m} \]

### Step 4: Write the general equation for a wave propagating along the negative x-direction
The general equation for a wave propagating in the negative x-direction is given by:
\[ y = A \sin(\omega t + kx) \]
Substituting the values of A, ω, and k:
\[ y = 0.02 \sin(1320\pi t + 4\pi x) \]

### Step 5: Simplify the equation
We can factor out \(2\pi\) from the sine function:
\[ y = 0.02 \sin(2\pi(660 t + 2x)) \]

### Final Equation
Thus, the equation of the wave is:
\[ y = 0.02 \sin(2\pi(660 t + 2x)) \]

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