Problem 14 - Algebra 2

Question

Sketch the graph of a sound wave with the given period and amplitude. period \(0.02,\) amplitude 4

Step-by-Step Solution

Answer

The graph of the sound wave with a given period of 0.02 and an amplitude of 4 will be a regular sinusoidal wave; it will reach 4 at one-quarter of 0.02, back to 0 at half of 0.02, go to -4 at three-quarters of 0.02 and then return to 0 at the end of the period 0.02.

Step 1: Sketch the basic wave

Start by drawing a basic wave graph or a sine graph that starts at the origin of the coordinate plane (0,0). Ensure that the wave goes up to the positive y-axis, comes back to the x-axis, goes down to the negative y-axis, and again comes back to the x-axis. This completes one cycle of the wave.

Step 2: Add the amplitude

Now, on the y-axis, mark the positive and negative limits to represent the amplitude of the wave. Since the amplitude for the wave here is 4, mark a point as 4 on the y-axis above the origin and mark another point as -4 on the y-axis below the origin. Draw horizontal lines that will serve as references from these marked points.

Step 3: Add the period

On the x-axis, identify a point that represents the ending of one period/cycle of the wave. The period given here is 0.02, so mark a point 0.02 on the x-axis starting from origin. This point marks the end of one period or one complete cycle of the wave.

Step 4: Complete the graph

Ensure that the wave reaches its maximum point 4 at one-quarter of the period, then back to zero at half the period, to its minimum point -4 at three-quarters of the period and then back to zero at the end of the period at 0.02. This completes the graph of the sound wave with the given period and amplitude. Repeat the cycle in both the positive and negative directions of the x-axis for a full view of the wave.

Key Concepts

Sine Functions

Sine functions are fundamental in representing sound waves and many other periodic phenomena. The sine function can be written as \( f(x) = A \sin(Bx) \), where \( A \) is amplitude and \( B \) affects the period. This function produces a smooth, continuous wave that oscillates symmetrically above and below the x-axis.

When visualizing sine waves in graphs, each complete oscillation is known as a cycle. The curve starts at the origin, rises to a peak, returns to the baseline, descends to a trough, and ascends back to the starting point.
Some key characteristics of sine waves include:

• Symmetry across the y-axis for a standard sine function.
• Wavelength, which measures the cycle length horizontally.
• Periodic nature, making it repeat indefinitely.

The sine function is versatile, capturing the essence of many waveforms seen in nature, including sound waves.

Wave Amplitude

Wave amplitude refers to the height of the wave's peaks. It indicates how strong or loud the sound is perceived. In mathematical terms, amplitude is the maximum distance from the wave's equilibrium point, or the x-axis.

For example, in the equation \( f(x) = 4 \sin(Bx) \), the amplitude is 4. This means the wave rises to +4 and descends to -4 on the y-axis.
Understanding amplitude is crucial in applications like sound engineering and audio processing. Larger amplitudes correspond to louder sounds, while smaller amplitudes represent softer sounds.
Additionally, a few tips on interpreting amplitude:

• Peak amplitude defines the extreme points of the wave.
• It stays constant in a pure tone.
• Does not affect the wave's frequency or period.

Thus, amplitude is a clear indicator of the wave's intensity without altering its periodic cycle.

Wave Period

Wave period defines how long it takes for one complete cycle of the wave. This measurement is along the x-axis. It determines how frequently the wave repeats itself over a given time.

In the equation of a sine function \( f(x) = A \sin(Bx) \), the wave period \( T \) is calculated as \( T = \frac{2\pi}{B} \). A larger \( B \) decreases the period, increasing frequency, while a smaller \( B \) results in the opposite.
For instance, if the period is 0.02, the wave repeats every 0.02 units on the x-axis. Knowing the wave period is essential in fields like acoustics, where wave timing is crucial.
When analyzing wave periods, consider:

• Frequency, which is the inverse of the period.
• Short periods indicate high-frequency waves.
• Longer periods show low-frequency waves.

A clear understanding of the wave period assists in grasping the repetitive nature of sound waves.