91Ó°ÊÓ

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y(x, t)=(5.60 \mathrm{~cm}) \sin [(0.0340 \mathrm{rad} / \mathrm{cm}) x] \sin [(50.0 \mathrm{rad} / \mathrm{s}) t],\) where the origin is at the left end of the string, the \(x\) -axis is along the string, and the \(y\) -axis is perpendicular to the string. (a) Draw a sketch that shows the standing-wave pattern. (b) Find the amplitude of the two traveling waves that make up this standing wave. (c) What is the length of the string? (d) Find the wavelength, frequency, period, and speed of the traveling waves. (e) Find the maximum transverse speed of a point on the string. (f) What would be the equation \(y(x, t)\) for this string if it were vibrating in its eighth harmonic?

Step by step solution

01

Identify Known Values

The given equation for the wave is \(y(x, t) = (5.60 cm) \sin[(0.0340 rad/cm)x] \sin[(50.0 rad/s)t]\). From this equation and the problem, it can be noted that the amplitude (A) is 5.60 cm, the angular wave number (\(k\)) is 0.0340 rad/cm, and the angular frequency (\(ω\)) is 50.0 rad/s.

02

Calculate Amplitude

For a standing wave, the amplitude is the same as the amplitude of the two identical traveling waves that make it up. Therefore, the amplitude of the traveling waves is also 5.60 cm.

03

Find Length of the String

In the third harmonic, the string length equals three half-wavelengths. Therefore, \(L=3λ/2\). However, since \(k = 2π/λ\), we find that \(λ = 2π/k\). Substituting this into the above equation gives \(L = 3*(2π/k)/2 = 3π/k\). Substituting the known value of k, we find that \(L = 3π/0.0340 rad/cm = 277 cm\).

04

Find Wavelength, Frequency, Period, and Speed

From before, we know that λ = 2π/k. Therefore, λ = 2π/0.0340 rad/cm = 585 cm. To find the frequency, note that ω = 2πν, and therefore, ν = ω/2π = 50.0 rad/s / 2π = 8 Hz. The period T is the reciprocal of the frequency, so T = 1/ν = 0.125 s. The wave speed v can be found from the relationship v = λν, giving v = (585 cm)(8 Hz) = 4680 cm/s.

05

Calculate Maximum Transverse Speed

The speed of a point on the string is v = ωA. Substituting the known values of ω and A, we find that v = (5.60 cm)(50.0 rad/s) = 280 cm/s.

06

Eighth Harmonic Equation

For the eight harmonic, y(x, t) is changed by replacing the wave number with eight times its value. Therefore, for the eight harmonic, \(y_8(x, t) = 5.60 cm \sin[8(0.0340 rad/cm)x] \sin[50.0 rad/s t]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standing Waves

Standing waves are a fascinating phenomenon that occurs when waves traveling in opposite directions interfere with each other in a fixed medium. In the context of a string tied at both ends, like a guitar string, standing waves form due to the reflection of waves at the rigid boundaries.
This results in specific points, called nodes, where there is no motion, and other points, called antinodes, where the motion is maximum.

• The wave 'stands still' rather than traveling, hence the name 'standing wave'.
• These waves can be visualized as patterns formed on the string when it vibrates in one of its harmonics.

In the given exercise, the string is oscillating in its third harmonic. This means there are three antinodes and nodes are positioned in between along the length of the string.
Standing wave patterns are essential in understanding how musical instruments produce sound, providing a natural resonance that amplifies the sound.

Harmonics

Harmonics are essentially the natural frequencies at which a system resonates. For a string fixed at both ends, these harmonics relate to the integer multiples of its fundamental frequency.

• The fundamental frequency (first harmonic) is the lowest frequency of vibration.
• Higher harmonics correspond to higher frequencies that are integer multiples of the fundamental.

In the exercise, the string is oscillating in its third harmonic, meaning it is vibrating at three times the fundamental frequency.
Each harmonic has its characteristic pattern with a different number of nodes and antinodes. For instance, the third harmonic has two additional nodes compared to the fundamental frequency.
Understanding harmonics is crucial for designing musical instruments, as they determine the pitch and timbre of the sound produced.

Wave Equation

The wave equation provided in the exercise, \(y(x, t) = (5.60 \text{ cm}) \sin[(0.0340 \text{ rad/cm}) x] \sin[(50.0 \text{ rad/s}) t]\), describes the displacement of the string at any point \(x\) and time \(t\). This equation is typical for a standing wave formed from two traveling waves of identical frequency and amplitude moving in opposite directions.

• It consists of two sine functions: one dependent on space \(x\) and the other on time \(t\).
• The amplitude \(A = 5.60 \text{ cm}\) indicates the maximum displacement from the rest position.

Breaking down the equation:

• The term \(0.0340 \text{ rad/cm}\) represents the wave number \(k\), which is related to the wavelength by \(k = \frac{2\pi}{\lambda}\).
• The term \(50.0 \text{ rad/s}\) represents the angular frequency \(\omega\), related to the frequency \(u\) by \(\omega = 2\piu\).

Solving this equation helps us understand the characteristics of the wave, such as wavelength, frequency, and speed.

Transverse Waves

Transverse waves are a type of wave where the displacement of the medium is perpendicular to the direction of wave propagation. In the context of the exercise, the wave on the string is a transverse wave because the string moves up and down while the wave travels along its length.
This type of wave is common in vibrating strings and electromagnetic waves.

• In a transverse wave on a string, peaks and troughs are created as the wave travels, showing motion perpendicular to the string's length.
• Unlike longitudinal waves, where the displacement occurs in the direction of travel (such as sound waves in air), transverse waves are easily visible, making them simpler to analyze.

Understanding transverse waves is vital in many fields of physics, including optics and acoustics, where the nature of wave propagation dictates how energy and information are transferred through a medium.