Q.In the given progressive wave $y = 5\sin(100\pi t - 0.4\pi x)$ where $y$ and $x$ are in m, $t$ is in s. What is the

The given equation describes a sinusoidal progressive wave, whose properties can be determined by comparing it with the standard form $y = A\sin(\omega t - kx)$.

1. Identify Parameters:

   Comparing the given wave equation $y = 5\sin(100\pi t - 0.4\pi x)$ with the standard form $y = A\sin(\omega t - kx)$, we identify:

   • Amplitude $A = 5$ m
   • Angular frequency $\omega = 100\pi$ rad/s
   • Wave number $k = 0.4\pi$ rad/m

2. Calculate Wavelength:

   The wave number $k$ is related to the wavelength $\lambda$ by the formula $k = \frac{2\pi}{\lambda}$.

   $\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.4\pi} = \frac{2}{0.4} = 5$ m.

3. Calculate Frequency:

   The angular frequency $\omega$ is related to the frequency $f$ by the formula $\omega = 2\pi f$.

   $f = \frac{\omega}{2\pi} = \frac{100\pi}{2\pi} = 50$ Hz.

4. Calculate Wave Velocity:

   The wave velocity $v$ can be calculated using $v = f\lambda$ or $v = \frac{\omega}{k}$.

   $v = (50 \text{ Hz})(5 \text{ m}) = 250$ m/s.

5. Calculate Particle Velocity Amplitude: …

When analyzing a progressive wave, its mathematical representation contains all the information about its physical properties. The key is to understand the standard form of a wave equation and then compare it with the given equation to extract the relevant parameters.

A general equation for a one-dimensional progressive harmonic wave traveling in the positive $x$-direction can be written as:

or

where:

• $y$ is the displacement of a particle from its equilibrium position at a given $x$ and $t$.
• $A$ is the amplitude, representing the maximum displacement of any particle from its equilibrium position.
• $k$ is the angular wave number (or propagation constant), related to the wavelength $\lambda$ by $k = \frac{2\pi}{\lambda}$.
• $\omega$ is the angular frequency, related to the frequency $f$ by $\omega = 2\pi f$.
• $t$ is time.
• $x$ is the position.
• $\phi$ is the initial phase constant.

The given wave equation is $y = 5\sin(100\pi t - 0.4\pi x)$. This equation matches the form $y(x, t) = A\sin(\omega t - kx)$, with the phase constant $\phi = 0$.

Let's extract each parameter by direct comparison and using the relevant definitions.

1. Identify Amplitude ($A$)

   The amplitude is the coefficient of the sine function in the wave equation.

   Comparing $y = 5\sin(100\pi t - 0.4\pi x)$ with $y = A\sin(\omega t - kx)$, we see that $A = 5$.

   Since $y$ is in meters, the amplitude is in meters.

2. Identify Angular Frequency ($\omega$) and Angular Wave Number ($k$)

   By comparing the terms inside the sine function:

   The coefficient of $t$ is the angular frequency $\omega$.

   So, $\omega = 100\pi$ rad/s.

   The coefficient of $x$ is the angular wave number $k$.

   So, $k = 0.4\pi$ rad/m.

3. Calculate Wavelength ($\lambda$)

   The angular wave number $k$ is defined as $k = \frac{2\pi}{\lambda}$. We can rearrange this to find the wavelength $\lambda$.

Substitute the value of $k$:

4. Calculate Frequency ($f$) The angular frequency $\omega$ is defined as $\omega = 2\pi f$. We can rearrange this to find the frequency $f$.

Substitute the value of $\omega$:

5. Calculate Wave Velocity ($v$) The wave velocity (or phase velocity) $v$ is the speed at which the wave propagates through the medium. It can be calculated using the relationship $v = \frac{\omega}{k}$ or $v = f\lambda$. Using $v = \frac{\omega}{k}$:

Alternatively, using $v = f\lambda$:

6. Calculate Particle Velocity Amplitude ($v_{p,max}$) …

Step 1: Compare $y=5\sin(100\pi t-0.4\pi x)$ with the standard form $y=A\sin(\omega t-kx)$: read off $A=5\ \text{m}$, $\omega=100\pi\ \text{rad/s}$, $k=0.4\pi\ \text{rad/m}$.

Step 2 (b): $\lambda = \dfrac{2\pi}{k} = \dfrac{2\pi}{0.4\pi} = 5\ \text{m}$.

Step 3 (c): $f = \dfrac{\omega}{2\pi} = \dfrac{100\pi}{2\pi} = 50\ \text{Hz}$.

Step 4 (d): $v = \dfrac{\omega}{k} = \dfrac{100\pi}{0.4\pi} = 250\ \text{m/s}$ (check: $f\lambda = 50\times5 = 250\ \text{m/s}$ ✓). …