The displacement of the particle at x = 0 of a stretched string carrying a wave in the positive x-direction is given by `f(t)=Asin(t/T)`. The wave speed is v. Write the wave equation.

Explanation

To derive the wave equation from the given displacement function, we can follow these steps:

### Step 1: Understand the given displacement function
The displacement of the particle at \( x = 0 \) is given by:
\[ f(t) = A \sin\left(\frac{t}{T}\right) \]
where \( A \) is the amplitude, \( t \) is the time, and \( T \) is the time period.

### Step 2: Identify the wave speed and its relationship with wavelength and frequency
The wave speed \( v \) is related to the wavelength \( \lambda \) and frequency \( \nu \) by the equation:
\[ v = \lambda \nu \]
Frequency \( \nu \) can also be expressed as:
\[ \nu = \frac{1}{T} \]
Thus, we can rewrite the wave speed equation as:
\[ v = \lambda \cdot \frac{1}{T} \]
Rearranging this gives:
\[ \lambda = vT \]

### Step 3: Write the general form of the wave equation
The general form of a wave traveling in the positive x-direction can be expressed as:
\[ y(x, t) = A \sin\left(\frac{2\pi}{\lambda} x - \omega t\right) \]
where \( \omega \) is the angular frequency given by:
\[ \omega = \frac{2\pi}{T} \]

### Step 4: Substitute the values into the wave equation
From the previous steps, we have:
- \( \lambda = vT \)
- \( \omega = \frac{2\pi}{T} \)

Substituting these into the wave equation gives:
\[ y(x, t) = A \sin\left(\frac{2\pi}{vT} x - \frac{2\pi}{T} t\right) \]

### Step 5: Simplify the wave equation
Since \( \frac{2\pi}{T} \) can be factored out, we rewrite the wave equation as:
\[ y(x, t) = A \sin\left(\frac{2\pi}{vT} x - \frac{2\pi}{T} t\right) \]
This can be expressed in terms of the wave speed \( v \):
\[ y(x, t) = A \sin\left(\frac{2\pi}{\lambda} x - \frac{2\pi}{T} t\right) \]
where \( \lambda = vT \).

### Final Wave Equation
Thus, the wave equation is:
\[ y(x, t) = A \sin\left(\frac{2\pi}{vT} x - \frac{2\pi}{T} t\right) \]