Problem 15 - Algebra 2

Question

Sketch the graph of a sound wave with the given period and amplitude. period \(0.005,\) amplitude 9

Step-by-Step Solution

Answer

To graph the sound wave with a period of \(0.005\) and amplitude of 9, you would plot a sine wave that oscillates between -9 and +9 on the y-axis and completes a full cycle by \(0.005\) on the x-axis.

Step 1: Draw the Axes with Appropriate Scales

Draw the x and y axes such that they can accommodate the values of the period and amplitude respectively. The x-axis represents time and should be labelled from 0 to a value slightly more than \(0.005\). The y-axis represents displacement and should be labelled from -9 to +9.

Step 2: Plot the Wave

Plot a sine wave starting from the origin (0,0). Let it peak at (0.0025, 9) which represents half of our period and the top of our wave. Then, let it return to the x-axis at (0.005, 0) which represents the end of our wave period. The wave should also pass through (0,0), and bottom out at (0.0025,-9) indicating the negative part of the wave's cycle.

Step 3: Mark the Key Points

Mark the peak and trough of the wave to highlight the amplitude, and mark the point where a full cycle is completed to show the period. These are the key points that characterize the wave. The full cycle is completed when the wave returns to its starting point, i.e., at a time value of \(0.005\) on the x-axis.

Key Concepts

Amplitude

The amplitude of a wave is an important concept in understanding sound waves, as it measures the height of the wave from its resting position. In simple terms, amplitude represents the maximum extent of a wave's displacement. For our sound wave, the amplitude is 9, meaning that the wave peaks at +9 and troughs at -9.

Amplitude is a critical factor in determining how "loud" a sound appears, as waves with higher amplitudes carry more energy and thus produce louder sounds. When graphing a sine wave such as our sound wave, you'd measure the amplitude vertically from the x-axis to the crest (top point) and the trough (lowest point).

Some useful points to remember about amplitude in wave graphs:

• Amplitude is always measured as an absolute value from the x-axis to the peak.
• It is independent of the length or speed of the wave.
• When graphing, label your y-axis to extend slightly beyond the amplitude values to ensure it is visually clear.

Period

The period of a wave describes the time it takes for one complete cycle of the wave to occur. For the sound wave in question, this period is given as 0.005. This means that the wave completes one full oscillation in 0.005 units of time.

To visualize the period on a graph:

• Begin plotting the wave at the origin and extend it to the right.
• It should reach one full cycle by the time it reaches the x-coordinate equal to the period.
• The cycle includes both going up to the peak and down to the trough before returning to the baseline.

Understanding the period is crucial in fields like music and acoustics, as it relates to the frequency (or pitch) of a sound. A shorter period means higher frequency, resulting in a higher pitched sound.

For practical considerations, always make sure your x-axis is properly labeled from 0 to just beyond the wave period, making it easy to identify where each cycle begins and ends.

Sine Wave

The sine wave is a fundamental concept when exploring wave patterns, especially in sound and other areas of physics. A sine wave is a smooth, periodic oscillation that graphically looks like a series of peaks and troughs, very much like "waves" in the ocean.

When applying the sine wave to the context of sound waves as in this exercise:

• We start at the origin (0,0) on the graph, representing the wave's start.
• The wave rises smoothly to the amplitude peak, here at (0.0025,9), which is halfway through the period, showcasing both the amplitude and period.
• Then, it descends back to the x-axis, crosses it, and continues to the negative amplitude before returning to complete the cycle at the given period of 0.005.

Recognizing the sinusoidal form of sound waves helps in understanding how different sounds can be represented graphically. Sine waves are often used because they are the simplest form of periodic wave and serve as a basis for more complex sounds and signals.