Question

The equation $\mathrm{y}=10\sin (\pi \mathrm{x}/4)\cos 10\pi t$ represents a stationary wave where $\mathrm{x}$ and $\mathrm{y}$ are in centimeter and $\mathrm{t}$ is in seconds. The amplitude of each component wave is $\dots \dots .$ (A) $5\mathrm{cm}$ (B) $10\mathrm{cm}$ (C) $20\mathrm{cm}$ (D) between $5\mathrm{cm}$ and $10\mathrm{cm}.$

Short answer

The amplitude of each component wave is 10 cm. Therefore, the correct answer is (B) $10\mathrm{cm}$.

Step-by-step solution

Identify the given stationary wave equation

The given stationary wave equation is $y=10\sin (\pi x/4)\cos (10\pi t)$.

Identify the component wave functions

The equation is a product of sine and cosine functions. The functions are: 1. $\sin (\pi x/4)$ 2. $\cos (10\pi t)$

Determine the amplitude of each component wave

Each component wave's amplitude is the maximum value of the corresponding sine or cosine function. For both sine and cosine functions, the maximum value is 1. Therefore, we need to find the coefficient that multiplies each function to determine the amplitude. For the first component wave (the sine function), the coefficient is 10. The amplitude of this component wave is: $A_1=10(1)=10\mathrm{cm}$ For the second component wave (the cosine function), the coefficient is also 10, but it is affected by the first component wave function. The amplitude of this component wave is: $A_2=10(1)=10\mathrm{cm}$

Determine the correct answer

The amplitude of each component wave is 10 cm. Therefore, the correct answer is (B) $10\mathrm{cm}$.

Key concepts

Wave Equation

The wave equation describes how waves propagate through a medium. In this case, we have a stationary wave represented by the equation $y=10\sin (\pi x/4)\cos (10\pi t)$. This is specific to scenarios where the wave does not appear to move, yet energy is transmitted. Stationary waves are formed by the superposition of two traveling waves moving in opposite directions with the same amplitude and frequency. Here, $x$ and $y$ are spatial dimensions measured in centimeters, and $t$ represents time in seconds. These components are involved in creating a wave that appears not to travel, evident through the multiplication of sine and cosine functions within the equation.

Amplitude

Amplitude refers to the maximum extent of a wave from its rest position. It is influenced by the coefficients in the wave equation. In our scenario, the amplitude is determined by the maximum values of the individual sine and cosine functions, each reaching up to 1. However, this maximum is scaled by the coefficient present in the wave equation. For the given equation $y=10\sin (\pi x/4)\cos (10\pi t)$, the coefficient is 10, meaning each component wave achieves an amplitude of:

• 10 cm for the sine component
• 10 cm for the cosine component

This scaling results in each part of the wave having equal amplitude, which is a pivotal feature in stationary waves.

Sine Function

The sine function plays a crucial role in wave equations, representing the variation of the wave over space. In the equation $\sin (\pi x/4)$, the sine function describes how the wave varies along the $x$ dimension. Here, $\pi x/4$ is the argument of the sine function, determining the number of wave cycles per unit length.Some essential properties of the sine function include:

• Repeats every $2\pi$, known as the period
• Maximum value of 1 and minimum value of -1
• Oscillates smoothly, creating the classic wave shape

Adjustment to the frequency and phase of the sine wave can topically affect the wave’s spatial characteristics when constructing stationary waves.

Cosine Function

The cosine function complements the sine in wave equations and describes how the wave varies over time. Within the given equation, the $\cos (10\pi t)$ portion explains the time variation. Analyzing this function reveals how often the wave reaches its peak during a cycle, defined here by the term $10\pi t$.Key characteristics of the cosine function include:

• Similar periodicity to the sine function, repeating every $2\pi$
• Starts at its maximum value when time is zero
• Oscillates smoothly, contributing to maintaining stationary characteristics in the wave

In sine and cosine functions, the multiplication by a factor (like 10 in this equation) indicates the amplitude of the oscillation, showing how large the wave variation is at its peak. This synergy between sine and cosine functions helps form the composite stationary wave.