The equation of a longitudinal wave is represented as `y= 20 cos pi (50 t-x)`. Its wavelength is

Options

A. 5 cm
B. 2 cm
C. 50 cm
D. 20 cm

Answer

To find the wavelength of the longitudinal wave represented by the equation \( y = 20 \cos(\pi (50t - x)) \), we can follow these steps:

### Step 1: Identify the standard form of the wave equation
The standard form of a longitudinal wave equation is given by:
\[ y = A \cos(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Rewrite the given equation
The given equation is:
\[ y = 20 \cos(\pi (50t - x)) \]
We can expand this:
\[ y = 20 \cos(50\pi t - \pi x) \]

### Step 3: Compare with the standard form
From the expanded equation, we can identify:
- \( \omega = 50\pi \)
- \( k = \pi \)

### Step 4: Relate wave number to wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
From our comparison, we have \( k = \pi \).

### Step 5: Solve for the wavelength
Substituting \( k = \pi \) into the wave number equation:
\[ \pi = \frac{2\pi}{\lambda} \]
To find \( \lambda \), we can rearrange the equation:
\[ \lambda = \frac{2\pi}{\pi} = 2 \]

### Step 6: State the units
Since the equation is in terms of \( x \) and \( y \) (which are typically in centimeters in such problems), the wavelength \( \lambda \) is:
\[ \lambda = 2 \text{ cm} \]

### Conclusion
The wavelength of the wave is \( 2 \text{ cm} \).

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