Question

Two waves are generated in a ripple tank. Suppose the height, in centimetres, above the surface of the water, of the waves can be modelled by \(f(x)=\sin x\) and \(g(x)=3 \sin x,\) where \(x\) is in radians. a) Graph \(f(x)\) and \(g(x)\) on the same set of coordinate axes. b) Use your graph to sketch the graph of \(h(x)=(f+g)(x)\) c) What is the maximum height of the resultant wave?

Short answer

The maximum height of the resultant wave is 4 cm.

Step-by-step solution

- Understanding Functions

Identify the given functions. Here, the functions are given by \[ f(x) = \sin x \] \[ g(x) = 3 \sin x \] These functions model the height of the waves above the water surface, where \( x \) is in radians.

- Plotting f(x)

Graph the function \( f(x) = \sin x \) on a set of coordinate axes. The sine function oscillates between -1 and 1 with a period of \( 2\pi \).

- Plotting g(x)

Graph the function \( g(x) = 3 \sin x \) on the same set of coordinate axes. This function also oscillates with a period of \( 2\pi \), but with amplitude 3, so it ranges from -3 to 3.

- Adding the Functions

To find the graph of \( h(x) = (f + g)(x) \), add the corresponding values of \( f(x) \) and \( g(x) \) at each point. Hence, \[ h(x) = f(x) + g(x) = \sin x + 3 \sin x = 4 \sin x \]

- Plotting h(x)

Graph \( h(x) = 4 \sin x \) on the same set of coordinate axes. This function oscillates with a period of \( 2\pi \) and an amplitude of 4, ranging from -4 to 4.

- Determine Maximum Height

The maximum value of \( h(x) = 4 \sin x \) occurs when \( \sin x \) is 1. Therefore, the maximum height of the resultant wave is 4 centimeters.

Key concepts

trigonometric functions

Trigonometric functions are fundamental in understanding wave patterns. They describe periodic oscillations, like the waves on a ripple tank.
This function is critical in modeling periodic motions because it repeats its values in regular intervals.
In our exercise, the functions given are \( f(x) = \sin x \) and \( g(x) = 3 \sin x \).
In both cases, sin x represents the vertical position of points on a wave at different x values (radians).

• The sine function ( \(\sin x \)) oscillates between -1 and 1 for all x values, which means it rises and falls in a predictable pattern.
• It's periodic with a fundamental period of \( 2\pi \), meaning every \( 2\pi \) units along the x-axis, the wave pattern repeats itself.

This oscillating characteristic is vital because it helps us predict wave behavior.

amplitude and period

Two key properties of trigonometric functions are amplitude and period. \(Amplitude \) is the height of the wave from its central axis, and \(period \) is the distance over which the wave's shape repeats.

• In \( f(x)=\sin x \), the amplitude is 1, meaning it varies between -1 and 1.
• The function repeats every \( 2\pi \) units along the x-axis, so its period is \( 2\pi \).
• For the function \( g(x) = 3\sin x \), the amplitude is 3, indicating it varies between -3 and 3.
• Like \( f(x) \), \( g(x) \) also repeats every \( 2\pi \) units, so its period is \( 2\pi \).

Understanding these properties helps us discern the wave's height and repetition rate.
For instance, \( g(x) \) has a higher wave due to its amplitude of 3, while \( f(x) \) has a lower amplitude, only reaching 1.

function addition

Function addition combines two functions to produce a new function. In our case, we are asked to find and graph \( h(x) = (f + g)(x) \).
Here's how it works:

• Evaluate \( f(x) = \sin x \) and \( g(x) = 3 \sin x \) at the same x values.
• Add their values together to obtain the new function \( h(x) \).
• Mathematically, \(h(x) = f(x) + g(x) = \sin x + 3 \sin x = 4 \sin x \).

This means that at any point on the x-axis, you add the sine values multiplied by their respective coefficients to get the result.
Plotting \( h(x) = 4 \sin x \) will show it oscillates between -4 and 4, with an amplitude of 4 and a period of \( 2\pi \).
Thus, the maximum height of the resultant wave is 4, occurring when the sine function reaches its peak value of 1.