The equation below represents a progressive wave `y=3 xx 10^(-7) sin (8500t-25x)`
where t is in seconds x and y in metres. Calculate the speed of the wave and the direction of propagation.

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the wave equation
The wave equation is given as:
\[ y = 3 \times 10^{-7} \sin(8500t - 25x) \]

### Step 2: Extract the angular frequency (ω) and wave number (k)
From the equation, we can identify:
- The angular frequency \( \omega = 8500 \, \text{rad/s} \)
- The wave number \( k = 25 \, \text{rad/m} \)

### Step 3: Use the relationship between wave speed (v), angular frequency (ω), and wave number (k)
The relationship between these quantities is given by the formula:
\[ v = \frac{\omega}{k} \]

### Step 4: Substitute the values of ω and k into the equation
Now, substituting the values we have:
\[ v = \frac{8500}{25} \]

### Step 5: Calculate the speed of the wave
Now, performing the division:
\[ v = 340 \, \text{m/s} \]

### Step 6: Determine the direction of propagation
In the wave equation, the term \( (8500t - 25x) \) indicates the direction of propagation. Since the equation has a negative sign between \( \omega t \) and \( kx \), the wave propagates in the positive x-direction.

### Final Answer
- Speed of the wave: \( 340 \, \text{m/s} \)
- Direction of propagation: Positive x-direction

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