A certain transverse wave is described by
`y(x, t)=(6.50 mm) cos 2pi((x)/(28.0 cm) - (t)/(0.0360 s)).`
Determine the wave's
(a) amplitude , (b) wavelength
( c ) frequency , (d) speed of propagation and
(e) direction of propagation.

Explanation

To solve the problem, we will analyze the given wave equation step by step and extract the required parameters.

The wave equation is given as:
\[ y(x, t) = (6.50 \, \text{mm}) \cos \left( 2\pi \left( \frac{x}{28.0 \, \text{cm}} - \frac{t}{0.0360 \, \text{s}} \right) \right) \]

### Step 1: Determine the Amplitude
The amplitude \( A \) is the coefficient in front of the cosine function in the wave equation.

From the equation:
\[ A = 6.50 \, \text{mm} \]

To convert this to meters:
\[ A = 6.50 \times 10^{-3} \, \text{m} \]

### Step 2: Determine the Wavelength
The wave number \( k \) is given by the term inside the cosine function related to \( x \):
\[ k = \frac{2\pi}{\lambda} \]

From the equation, we can identify:
\[ k = \frac{2\pi}{28.0 \, \text{cm}} \]

To find the wavelength \( \lambda \):
\[ \lambda = 28.0 \, \text{cm} \]

### Step 3: Determine the Frequency
The angular frequency \( \omega \) is given by the term inside the cosine function related to \( t \):
\[ \omega = \frac{2\pi}{T} \]

From the equation, we can identify:
\[ \omega = \frac{2\pi}{0.0360 \, \text{s}} \]

To find the frequency \( f \):
Using the relationship \( \omega = 2\pi f \):
\[ f = \frac{\omega}{2\pi} = \frac{1}{0.0360 \, \text{s}} \]

Calculating this gives:
\[ f \approx 27.78 \, \text{Hz} \]

### Step 4: Determine the Speed of Propagation
The speed of propagation \( v \) can be calculated using the formula:
\[ v = f \lambda \]

Substituting the values we found:
\[ v = (27.78 \, \text{Hz}) \times (28.0 \, \text{cm}) \]

Converting \( 28.0 \, \text{cm} \) to meters:
\[ v = (27.78 \, \text{Hz}) \times (0.28 \, \text{m}) \]

Calculating this gives:
\[ v \approx 7.78 \, \text{m/s} \]

### Step 5: Determine the Direction of Propagation
In the wave equation, the term \( -\frac{t}{0.0360 \, \text{s}} \) indicates that the wave is traveling in the positive x-direction.

### Summary of Results
(a) Amplitude: \( 6.50 \, \text{mm} = 6.50 \times 10^{-3} \, \text{m} \)
(b) Wavelength: \( 28.0 \, \text{cm} = 0.28 \, \text{m} \)
(c) Frequency: \( 27.78 \, \text{Hz} \)
(d) Speed of propagation: \( 7.78 \, \text{m/s} \)
(e) Direction of propagation: Positive x-direction