A wave travelling along the x-axis is described by the equation y(x,t) = 0.005 cos(alpha x - beta t)

Q: Why did we use cosine instead of sine for the standard equation?

Both sine and cosine represent valid harmonic waves. They only differ by a phase shift of $\frac{\pi}{2}$. Since the given equation uses cosine, comparing it with a cosine standard equation is the most direct approach.

Q: What does the negative sign in $(\alpha x - \beta t)$ indicate?

The negative sign indicates that the wave is travelling in the positive x-direction. If it were a positive sign, the wave would be travelling in the negative x-direction.

Q: What are the units of $\alpha$ and $\beta$?

$\alpha$ represents the wave number $k$, so its unit is $\text{m}^{-1}$ (or $\text{rad/m}$). $\beta$ represents the angular frequency $\omega$, so its unit is $\text{s}^{-1}$ (or $\text{rad/s}$).

Q: Does the amplitude $0.005$ affect the values of $\alpha$ and $\beta$?

No, the amplitude only determines the maximum displacement of the wave. It is completely independent of the spatial ($\alpha$) and temporal ($\beta$) frequencies.

Q: Can we find the wave speed from this information?

Yes! The wave speed $v$ is given by $v = \frac{\omega}{k} = \frac{\beta}{\alpha}$. You can also calculate it as $v = \frac{\lambda}{T}$.

Question

A wave travelling along the x-axis is described by the equation $y(x,t) = 0.005 \cos(\alpha x - \beta t)$. If the wavelength and the time period of the wave are $0.08	ext{ m}$ and $2.0	ext{ s}$ respectively, then $\alpha$ and $\beta$ in appropriate units are

Options:

(A) $\alpha = 25.00 \pi, \beta = \pi$

(B) $\alpha = \frac{0.08}{\pi}, \beta = \frac{2.0}{\pi}$

(C) $\alpha = \frac{0.04}{\pi}, \beta = \frac{1.0}{\pi}$

(D) $\alpha = 12.50 \pi, \beta = \frac{\pi}{2.0}$

Sigma Percentile · Accepted Answer

### Decoding the Wave Equation

Imagine you are watching a ripple travel across a pond. To describe this motion mathematically, physicists use the standard wave equation. Let's break down how we can extract meaningful physical quantities from a given mathematical expression.

### The Master Equation

The standard equation for a harmonic wave travelling in the positive x-direction is given by:

$$y(x,t) = A \cos(kx - \omega t)$$

Here, $A$ is the amplitude, $k$ is the wave number (which tells us about the spatial frequency), and $\omega$ is the angular frequency (which tells us about the temporal frequency).

Now, let's look at the equation provided in our problem:

$$y(x,t) = 0.005 \cos(\alpha x - \beta t)$$

By simply comparing our given equation with the standard form, we can establish a direct mapping:

**$\alpha$ corresponds to the wave number $k$**
**$\beta$ corresponds to the angular frequency $\omega$**

### Calculating the Spatial Component ($\alpha$)

The wave number $k$ is intimately connected to the wavelength $\lambda$ through the relation:

$$k = \frac{2\pi}{\lambda}$$

We are given that the wavelength $\lambda = 0.08	ext{ m}$. Substituting this value into our formula gives:

$$\alpha = \frac{2\pi}{0.08}$$

$$\alpha = 25\pi 	ext{ m}^{-1}$$

### Calculating the Temporal Component ($\beta$)

Similarly, the angular frequency $\omega$ is related to the time period $T$ by the equation:

$$\omega = \frac{2\pi}{T}$$

We are given that the time period $T = 2.0	ext{ s}$. Substituting this value yields:

$$\beta = \frac{2\pi}{2.0}$$

$$\beta = \pi 	ext{ s}^{-1}$$

### The Final Result

We have successfully decoded the constants from the wave equation. The values are **$\alpha = 25.00 \pi$** and **$\beta = \pi$**. This perfectly matches option (a).

This problem beautifully illustrates how abstract mathematical constants in a wave equation are directly tied to tangible physical properties like wavelength and time period.

Animated Solution for Physics - Waves: A wave travelling along the x-axis is described by the equation $y (x, t) = 0.005 cos (α x - β t)$ . If the wavelength and the time period of the wave are $0.08 m$ and $2.0 s$ respectively, then $α$ and $β$ in appropriate units are

Select Answer:

Visualized Solution

Standard Wave Equation

Calculating $α$

Calculating $β$

The Sigma Insight: Wave Equation and Wave Speed

Decoding the Wave Equation

The Master Equation

Calculating the Spatial Component ( $α$ )

Calculating the Temporal Component ( $β$ )

The Final Result

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Visualized Solution

Standard Wave Equation

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β

The Sigma Insight: Wave Equation and Wave Speed

Decoding the Wave Equation

The Master Equation

Calculating the Spatial Component ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α)

Calculating the Temporal Component ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β)