Question 77: A wave is represented by the equation; y = A sin(10 pi x + 15 pi t + pi/3) where, x is

Q: Why does having the same sign for $kx$ and $\omega t$ mean the wave is travelling in the negative $x$-direction? It feels counter-intuitive!

Think of a constant phase point on the wave: $10\pi x + 15\pi t = \text{constant}$. As time $t$ increases, for the sum to remain constant, the position $x$ must decrease (become more negative). Thus, the wave profile shifts to the left, which is the negative $x$-direction.

Q: Does the phase constant $\pi/3$ affect the wave speed or wavelength?

Not at all! The phase constant $\phi = \pi/3$ only represents the initial shift of the wave at $t=0$. It has absolutely no influence on the spatial frequency (wavelength) or temporal frequency (wave speed).

Q: Can I use $v = f\lambda$ instead of $v = \omega/k$?

Yes, absolutely! Since $\omega = 2\pi f$ and $k = \frac{2\pi}{\lambda}$, substituting these into $v = \frac{\omega}{k}$ gives $v = \frac{2\pi f}{2\pi/\lambda} = f\lambda$. Both formulas are completely equivalent and will yield the exact same result.

Q: What is the difference between particle velocity and wave velocity?

This is a classic point of confusion! Wave velocity ($v = \omega/k$) is the speed at which the wave profile (or phase) propagates through the medium. Particle velocity is the speed at which individual particles of the medium oscillate about their equilibrium positions, given by $v_p = \frac{\partial y}{\partial t}$.

Q: What would happen if the wave equation used a cosine function instead of a sine function?

The wave speed, wavelength, and direction of propagation would remain completely identical. A cosine wave is simply a sine wave shifted by a phase of $\pi/2$. The physics of propagation depends entirely on the coefficients of $x$ and $t$.

Question

A wave is represented by the equation;

$$y = A \sin (10 \pi x + 15 \pi t + \pi / 3)$$

where, $x$ is in metre and $t$ is in second. The expression represents

Options:

(A) a wave travelling in the positive $x$-direction with a velocity $1.5	ext{ m/s}$

(B) a wave travelling in the negative $x$-direction with a velocity $1.5	ext{ m/s}$

(C) a wave travelling in the negative $x$-direction with a wavelength $0.2	ext{ m}$

(D) a wave travelling in the positive $x$-direction with a wavelength $0.2	ext{ m}$

Sigma Percentile · Accepted Answer

### The Magic of Wave Equations

Imagine standing on the shore of a calm lake, watching ripples spread outward after a pebble is dropped. The rhythmic rise and fall of the water surface is a beautiful physical manifestation of a wave. In physics, we capture this dynamic beauty using a mathematical tool called the **wave equation**.

At first glance, a wave equation like $y = A \sin(10\pi x + 15\pi t + \pi/3)$ might look like a daunting wall of symbols. But once you learn to decode its anatomy, it reveals a thrilling story of motion, energy, and geometry. Let us embark on a journey to dissect this equation and uncover the secrets of the wave it represents.

### Decoding the Anatomy of a Wave Equation

To understand any progressive wave, we must compare it to its universal template:
$$y = A \sin(kx \pm \omega t + \phi)$$

Each term in this template has a specific physical meaning:
- $A$ is the **amplitude**, representing the maximum displacement of the particles from their equilibrium position.
- $k$ is the **wave number**, which measures how rapidly the wave oscillates in space (spatial frequency).
- $\omega$ is the **angular frequency**, which measures how rapidly the wave oscillates in time (temporal frequency).
- $\phi$ is the **initial phase**, which determines the starting position of the wave at $t = 0$.

By aligning our given equation with this template, we can read its physical properties directly.

### The Sign Language of Waves: Direction of Propagation

One of the most common traps in wave mechanics is determining the direction of propagation. Many students intuitively assume that a plus sign means motion in the positive direction, but the reality is exactly the opposite!

Let us look at the phase term: $(10\pi x + 15\pi t + \pi/3)$. For a specific point on the wave (like a peak or a trough) to maintain its shape, its phase must remain constant:
$$10\pi x + 15\pi t + \pi/3 = 	ext{constant}$$

Now, let us think dynamically. As time $t$ increases, the term $15\pi t$ grows larger. For the sum to remain constant, the term $10\pi x$ must decrease. This means the position $x$ must move in the negative direction!

Therefore, whenever the coefficients of $x$ and $t$ have the **same sign** (both positive or both negative), the wave travels in the **negative $x$-direction**. Conversely, if they have **opposite signs**, the wave travels in the **positive $x$-direction**. Since both $10\pi x$ and $15\pi t$ are positive in our equation, our wave is travelling to the left, in the negative $x$-direction.

### Extracting the DNA of the Wave

Now that we have established the direction of motion, let us extract the numerical values of $k$ and $\omega$:
- Wave number: $k = 10\pi 	ext{ rad/m}$
- Angular frequency: $\omega = 15\pi 	ext{ rad/s}$

These two parameters are the fundamental genetic code of the wave. From them, we can calculate the physical dimensions of the wave: its wavelength and its speed.

### Calculating the Wavelength ($\lambda$)

Wavelength is the physical distance between two consecutive identical points on a wave, such as two adjacent peaks. It is inversely proportional to the wave number $k$ through the relation:
$$k = \frac{2\pi}{\lambda}$$

Rearranging this formula to solve for $\lambda$:
$$\lambda = \frac{2\pi}{k}$$

Substituting our value of $k = 10\pi$:
$$\lambda = \frac{2\pi}{10\pi} = 0.2 	ext{ m}$$

The $\pi$ terms cancel out beautifully, leaving us with a clean, elegant wavelength of $0.2	ext{ m}$.

### Calculating the Wave Velocity ($v$)

The speed of a wave is the rate at which the wave profile propagates through the medium. It is determined by the ratio of how fast the wave oscillates in time ($\omega$) to how fast it oscillates in space ($k$):
$$v = \frac{\omega}{k}$$

Substituting our values of $\omega = 15\pi$ and $k = 10\pi$:
$$v = \frac{15\pi}{10\pi} = 1.5 	ext{ m/s}$$

Once again, the $\pi$ terms cancel out, giving us a wave speed of $1.5	ext{ m/s}$. Since the wave is travelling in the negative $x$-direction, its velocity is $-1.5\hat{i}	ext{ m/s}$.

### Celebrating the Elegance of Physics

By systematically breaking down the wave equation, we have solved the mystery. The wave travels in the **negative $x$-direction** with a velocity of **$1.5	ext{ m/s}$** and has a wavelength of **$0.2	ext{ m}$**.

This perfectly matches options **(b)** and **(c)**. The beauty of physics lies in how a simple mathematical expression can encapsulate a complete physical reality. The next time you see a wave equation, do not see it as a formula—see it as a living, breathing description of motion!

Animated Solution for Physics - Waves: A wave is represented by the equation; $y = A sin (10 π x + 15 π t + π /3)$ where, $x$ is in metre and $t$ is in second. The expression represents

Select Answer:

Visualized Solution

Visualizing the Wave Equation

The Standard Progressive Wave Equation

Determining the Direction of Motion

Extracting Wave Parameters

The Wavelength Formula

Solving for $λ$

The Wave Velocity Formula

Solving for $v$

Selecting the Correct Options

Exploring Wave Phase Shifts

The Sigma Insight: Wave Equation and Wave Speed

The Magic of Wave Equations

Decoding the Anatomy of a Wave Equation

The Sign Language of Waves

Extracting the DNA of the Wave

Calculating the Wavelength ( $λ$ )

Calculating the Wave Velocity ( $v$ )

Celebrating the Elegance of Physics

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Animated Solution for Physics - Waves: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A wave is represented by the equation; y=Asin(10πx+15πt+π/3) where, x is in metre and t is in second. The expression represents