Question

A wave travelling along a string is described by the equation \(\mathrm{y}=\mathrm{A} \sin (\omega \mathrm{t}-\mathrm{kx}) .\) The maximum particle velocity is \(\ldots \ldots\) (A) A\omega (B) \(\omega / \mathrm{k}\) (C) \(\mathrm{d} \omega / \mathrm{dk}\) (D) \(\mathrm{x}\)

Short answer

The maximum particle velocity can be found by deriving the wave equation with respect to time: \(\frac{dy}{dt} = A \cos(\omega t - kx) \cdot \omega\). The maximum value occurs when \(\cos(\omega t - kx) = 1\), resulting in a maximum particle velocity of \(A\omega\), which corresponds to option (A).

Step-by-step solution

Derive the wave equation with respect to time

We will derive the wave equation, \(y=A \sin(\omega t-kx)\), with respect to time (t) to get the particle velocity, as the velocity represents the rate of change in vertical displacement (y) with respect to time: \[\frac{dy}{dt} = A \frac{d}{dt} [\sin(\omega t - kx)]\]

Apply the chain rule

Here, we will apply the chain rule because we have a composition of functions, i.e., a function of \(\omega t - kx\): \[\frac{d}{dt} [\sin(\omega t - kx)] = \cos (\omega t - kx) \cdot \frac{d}{dt} [\omega t - kx]\]

Derive \(\omega t - kx\) with respect to time

Derive the inner function \((\omega t - kx)\) with respect to time (t): \[\frac{d}{dt} [\omega t - kx] = \omega\]

Multiply to get the particle velocity

Multiply the results from step 2 and step 3 to get the particle velocity: \[\frac{dy}{dt} = A \cos(\omega t - kx) \cdot \omega\]

Find the maximum particle velocity

To find the maximum particle velocity, we need to find the maximum value of \(\frac{dy}{dt}\). We know that the maximum value of \(\cos(\omega t - kx)\) is 1. Therefore, the maximum particle velocity will be when \(\cos(\omega t - kx) = 1\): \[\frac{dy}{dt} = A \cdot 1 \cdot \omega\] Therefore, the maximum particle velocity is \(A\omega\), which corresponds with option (A).

Key concepts

Sinusoidal Waves

Sinusoidal waves are fundamental to understanding wave motion. They are one of the simplest types of periodic waves described by sine or cosine functions. These waves are characterized by:

• Amplitude (A): The peak value of the wave, determining its height.
• Angular Frequency (\omega): Related to how fast the wave oscillates per unit time.
• Wave Number (k): Tells how many oscillations occur per unit length.

In mathematical terms, a sinusoidal wave travelling in one dimension on a medium is often expressed as \(y = A \sin(\omega t - kx)\), where \(t\) is the time variable and \(x\) represents the position. Understanding this equation helps unveil the wave's behavior in terms of phase and amplitude changes as it moves through space. Sinusoidal waves are prevalent in many physical phenomena, including sound waves, light waves, and water waves.

Particle Velocity

Particle velocity in the context of wave motion refers to how fast a particle of the medium moves as the wave passes through it. It's different from wave velocity, which is about how fast the disturbance travels along the medium.
To find the particle velocity, you must derive the position function of the wave with respect to time. Given a wave function \(y = A \sin(\omega t - kx)\), the particle velocity \(v_p\) is found by calculating \(\frac{dy}{dt}\).
This derivative represents the rate of change of vertical displacement, showing how amplitude at a point changes over time. Calculating particle velocity is crucial in fields like acoustics, where understanding the movement of particles helps assess sound propagation and intensity.

Chain Rule in Calculus

The chain rule is a fundamental calculus tool crucial for deriving composite functions. When dealing with sinusoidal waves, especially when finding particle velocities, the chain rule plays a vital role. Let's break it down:

• If you have a composite function like \(\sin(\omega t - kx)\), you can't directly differentiate it without considering its internal components.
• By using the chain rule, you multiply the derivative of the outer function \(\sin(u)\) with respect to its argument \(u\), with the derivative of the inner function \(\omega t - kx\) with respect to \(t\).

Performing the calculation: \[\frac{d}{dt} [\sin(\omega t - kx)] = \cos(\omega t - kx) \cdot \frac{d}{dt}[\omega t - kx]\]This shows how the chain rule helps in breaking down complex wave equations into manageable components. Proper use of the chain rule makes analyzing changes in wave phenomena both simpler and more reliable.

Maximum Value of Trigonometric Functions

Trigonometric functions, like sine and cosine, have known maximum values, which greatly simplify calculations in wave physics. Understanding these maxima is essential.

• For \(\sin\) and \(\cos\), the maximum value is 1 and the minimum is -1.
• This characteristic is derived from the unit circle, where the trigonometric functions represent coordinates on the circle.

In the context of calculating maximum particle velocity, knowing that \(\cos(\omega t - kx)\) can at most equal 1 helps. This is because the term \(A \cos(\omega t - kx) \omega\) reaches its peak when \(\cos\) attains its maximum. Recognizing trigonometric function limits lets you accurately compute extrema in a variety of problems involving oscillations and periodic movements. This knowledge is key when looking for maximum effects, such as top speeds or peak forces in oscillatory systems.