Obtain the equation of a harmonic wave travelling in the negative x-direction with the characteristics given below. Amplitude is 4.5 cm, wavelength is 20 cm, wåve velocity is `40 ms^(-1)?`

Answer

To obtain the equation of a harmonic wave traveling in the negative x-direction with the given characteristics, we will follow these steps:

### Step 1: Identify the wave equation
The general equation for a wave traveling in the negative x-direction is given by:
\[ y = A \sin(\omega t + kx) \]
where:
- \( y \) is the displacement,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the angular wave number,
- \( t \) is the time,
- \( x \) is the position.

### Step 2: Determine the amplitude
From the question, the amplitude \( A \) is given as:
\[ A = 4.5 \text{ cm} \]

### Step 3: Calculate the angular wave number \( k \)
The angular wave number \( k \) can be calculated using the formula:
\[ k = \frac{2\pi}{\lambda} \]
where \( \lambda \) is the wavelength. Given that the wavelength \( \lambda \) is 20 cm, we convert it to meters:
\[ \lambda = 20 \text{ cm} = 0.2 \text{ m} \]
Now substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{0.2} = 10\pi \text{ m}^{-1} \]

### Step 4: Calculate the angular frequency \( \omega \)
We know the relationship between wave velocity \( v \), angular frequency \( \omega \), and angular wave number \( k \):
\[ v = \frac{\omega}{k} \]
Rearranging this gives:
\[ \omega = v \cdot k \]
Given that the wave velocity \( v \) is 40 m/s, we can substitute the values:
\[ \omega = 40 \cdot 10\pi = 400\pi \text{ rad/s} \]

### Step 5: Write the wave equation
Now we have all the necessary parameters:
- Amplitude \( A = 4.5 \text{ cm} \)
- Angular frequency \( \omega = 400\pi \text{ rad/s} \)
- Angular wave number \( k = 10\pi \text{ m}^{-1} \)

Substituting these values into the wave equation:
\[ y = 4.5 \sin(400\pi t + 10\pi x) \]

### Final Equation
Thus, the equation of the harmonic wave traveling in the negative x-direction is:
\[ y = 4.5 \sin(400\pi t + 10\pi x) \]

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