Question

MODELING WITH MATHEMATICS A middle-A tuning fork vibrates with a frequency \(f\) of 440 hertz (cycles per second). You strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. Write and graph a sine model that gives the pressure \(P\) as a function of the time \(t\) (in seconds).

Short answer

The correct model for the pressure as a function of time is given by the equation \(P(t) = 5\sin(440t)\).

Step-by-step solution

Understand the Situation

A tuning fork vibrating forms a sinusoidal wave with a frequency of 440 Hz and a maximum pressure of 5 Pa. Pressure output is sinusoidal because it's at a maximum when the tuning fork tines are at their furthest extent (greatest pressure), drops to zero (no pressure) when passing through the equilibrium position, and goes negative (a vacuum effect) when at the other extreme.

Formulate the Sine Function

In general, a sine wave model can be expressed as \(y = A\sin(B(x - C)) + D\), where:\n\n- \(A\) is the amplitude (maximum pressure in the case of the tuning fork application)\n- \(B\) is the frequency\n- \(C\) is the horizontal shift, as no phase shift is given here, therefore \(C = 0\)\n- \(D\) is the vertical shift (mean value around which the function oscillates, here \(D = 0\) as there was no shift given)\n\nGiven that the frequency is 440 Hz and the maximum pressure is 5 Pascals, the sine function becomes \(P(t) = 5\sin(440t)\). This function will give the pressure as a function of time.

Graph the Function

Graph \(y = 5\sin(440t)\). On the x-axis, place the time \(t\) (in seconds), while on the y-axis, place the pressure \(P\). Observe that the wave has a frequency of 440 hertz and an amplitude of 5, meaning it oscillates between -5 and 5 at a very high rate, 440 times per second to be exact.

Key concepts

Frequency of a Wave

The frequency of a wave is a fundamental characteristic that describes how often the wave oscillates or cycles over a specific period. Specifically, the frequency (often denoted as \( f \)) refers to the number of cycles that occur in one second, which is measured in hertz (Hz). It's crucial to understand that a higher frequency means more cycles per second, leading to a wave that seems to vibrate very quickly. For our middle-A tuning fork vibrating at 440 Hz, this translates to 440 complete sinusoidal cycles every second.

In the context of our exercise, frequency determines how 'tight' the pressure waves are. Imagine the pressure changing 440 times between its peak and trough within a single second; that's the effect of the frequency. When visualizing the sine graph, the frequency will dictate the horizontal distance between peaks and troughs of the wave, with higher frequencies leading to a more closely packed wave pattern. This concept is essential for understanding and modeling waves in various scientific and engineering fields, including acoustics, radio communications, and electronics.

Sinusoidal Wave

A sinusoidal wave is the graph of a sine function and is a smooth, periodic oscillation that is fundamental to understanding various types of wave phenomena. It is characterized by a smooth, repetitive rise and fall which looks like peaks and valleys on a graph. The sine function, described mathematically as \(y = A \sin(B(x - C)) + D\), models this behavior perfectly. It is used in various fields, from the study of sound, as in our tuning fork example, to alternating currents in electrical engineering.

The sine function consists of four main parameters: amplitude (\(A\)), which measures the height of the peaks; frequency (\(B\)), which determines how close together these peaks are; phase shift (\(C\)), which indicates whether the wave starts at a point other than the origin; and vertical shift (\(D\)), which moves the entire wave up or down. In our example, the tuning fork generating a pressure wave can be modeled by the sine function \(P(t) = 5\sin(440t)\), representing the sinusoidal oscillations of pressure over time.

Amplitude of a Wave

The amplitude of a wave refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium (middle) to the peak (top) or trough (bottom) of the wave. It is often denoted by \(A\) in the sinusoidal wave equation and represents how far the wave rises above or falls below its resting position. The amplitude is related directly to the energy and intensity of the wave: the higher the amplitude, the more energy is carried and the stronger the wave's effects.

In our exercise, the amplitude is 5 pascals, suggesting that the pressure wave from the tuning fork reaches a maximum pressure change of 5 pascals above and below the equilibrium pressure of the medium. This means that the total variation in pressure (peak to trough) is 10 pascals. In a graphical representation, the amplitude dictates the vertical stretch of the wave, which is the distance from the center line to the peak or trough. For the middle-A tuning fork, the amplitude indicates how strong the sound waves are, correlating to how loud the sound is perceived.