**Here, $x$ and $\Psi$ are measured in meters while $t$ is in seconds. Carefully study this wave equation and calculate the following quantities:**

\[\boldsymbol{ \Psi(x,t) = 4.8 cos ( 1.2x – 8.2t + 0.54 ) }\]

**– Frequency ( in hertz )**

**– Wavelength ( in meters )**

**– Wave speed ( in meters per second )**

**– Phase angle ( in radians )**

The aim of this question is to develop an understanding of the **traveling wave equation**.

To solve this question, we **simply compare** the given equation with the **standard wave equation** and then find the necessary parameters as given below:

\[ \Psi(x,t) = A cos ( k x – \omega t + \phi ) \]

Then we simply find **wavelength, speed and frequency** by following these formulas:

\[ f = \frac{ \omega }{ 2 \pi } \]

\[ \lambda = \frac{ 2 \pi }{ k } \]

\[ v = f \cdot \lambda \]

## Expert Answer

**Step 1:** Given the function:

\[ \Psi(x,t) = 4.8 \ cos ( 1.2x \ – \ 8.2t \ + \ 0.54 ) \]

The standard wave equation is given by:

\[ \Psi(x,t) = A \ cos ( k x \ – \ \omega t \ + \ \phi ) \]

**Comparing** the give equation with the **standard equation**, we can see that:

\[ A = 4.8 \]

\[ k = 1.2 \]

\[ \omega = 8.2 \ \frac{rad}{sec} \]

**\[ \phi = 0.54 \ rad \]**

**Step 2:** Calculating **Frequency:**

\[ f = \frac{ \omega }{ 2 \pi } \]

\[ f = \dfrac{ 8.2 \ \frac{rad}{sec} }{ 2 \pi \ rad} \]

**\[ f = 0.023 \ sec^{-1} \]**

**Step 3:** Calculating **Wavelength:**

\[ \lambda = \frac{ 2 \pi }{ k } \]

\[ \lambda = \frac{ 2 \pi }{ 1.2 } \]

\[ \lambda = 300 \ meter \]

**Step 4**: Calculating **Wave Speed:**

\[ v = f \cdot \lambda \]

\[ v = ( 0.023 \ sec^{-1}) ( 300 \ meter ) \]

**\[ v = 6.9 \ \frac{meter}{sec} \]**

## Numerical Result

For the given wave equation:

– Frequency ( in hertz ) **$ \boldsymbol{ f = 0.023 \ sec^{-1} }$**

– Wavelength ( in meters ) **$ \boldsymbol{ \lambda = 300 \ meter }$**

– Wave speed ( in meters per second ) **$ \boldsymbol{ v = 6.9 \ \frac{meter}{sec} }$**

– Phase angle ( in radians ) **$ \boldsymbol{ \phi = 0.54 \ rad }$**

## Example

Find **Frequency** (in hertz), **Wavelength** (in meters), **Wave speed** (in meters per second) and **Phase angle** (in radians) for the following wave equation:

\[ \Psi(x,t) = 10 cos ( x – t + \pi ) \]

**Comparing** with the **standard equation**, we can see that:

\[ A = 10 , \ k = 1, \ \omega = 1 \frac{rad}{sec}, \ \phi = \pi \ rad \]

Calculating **Frequency:**

\[ f = \frac{ \omega }{ 2 \pi } = \dfrac{ 1 \ \frac{rad}{sec} }{ 2 \pi \ rad} = \frac{1}{ 2 \pi } \ sec^{-1} \]

Calculating **Wavelength:**

\[ \lambda = \frac{ 2 \pi }{ k } = \frac{ 2 \pi }{ 1 } = 2 \pi \ meter \]

Calculating **Wave Speed:**

\[ v = f \cdot \lambda = ( \frac{1}{ 2 \pi } sec^{-1}) ( 2 \pi meter ) = 1 \ \frac{m}{s} \]