Given `y=0.8 sin 6pi [t+x/(40)]` metre. Calculate the wavelength and velocity of the wave represented by this equation.

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the wave equation
The given wave equation is:
\[ y = 0.8 \sin\left(6\pi t + \frac{x}{40}\right) \]

### Step 2: Rewrite the wave equation in standard form
We can rewrite the equation in the standard form of a wave:
\[ y = a \sin(\omega t + kx) \]
where:
- \( a \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

From the given equation, we can see:
- \( \omega = 6\pi \) (from the coefficient of \( t \)),
- \( k = \frac{6\pi}{40} = \frac{3\pi}{20} \) (from the coefficient of \( x \)).

### Step 3: Calculate the wavelength
The relationship between the wave number \( k \) and the wavelength \( \lambda \) is given by:
\[ k = \frac{2\pi}{\lambda} \]

To find the wavelength, we rearrange this equation:
\[ \lambda = \frac{2\pi}{k} \]

Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{\frac{3\pi}{20}} \]

### Step 4: Simplify the expression for wavelength
Now, simplify the expression:
\[ \lambda = \frac{2\pi \cdot 20}{3\pi} = \frac{40}{3} \text{ meters} \]

### Step 5: Calculate the velocity of the wave
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]

Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{6\pi}{\frac{3\pi}{20}} \]

### Step 6: Simplify the expression for velocity
Now, simplify the expression:
\[ v = 6\pi \cdot \frac{20}{3\pi} = \frac{120}{3} = 40 \text{ meters/second} \]

### Final Results
- Wavelength \( \lambda = \frac{40}{3} \) meters
- Velocity \( v = 40 \) meters/second

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