Question

a) Geben Sie für die Saite der Gitarre mit der Länge \(L=80 \mathrm{~cm}\) und der Wellengeschwindigkeit \(100 \mathrm{~m} / \mathrm{s}\) die Gleichung für die Grundschwingung und die dritte Oberschwingung an. Die Amplitude der Grundschwingung beträgt \(2 \mathrm{~cm}\), die der 3 . Oberschwingung \(1 \mathrm{~cm}\). b) An welchen Stellen befinden sich Knoten? c) An welchen Stellen befinden sich Schwingungsbäuche?

Short answer

The frequency of the fundamental mode is 62.5 Hz, and of the third harmonic is 187.5 Hz. The equations are \(y(x, t) = 0.02 \sin(\pi x)\cos(62.5t)\) for the fundamental mode, and \(y(x, t) = 0.01 \sin(3\pi x)\cos(187.5t)\) for the third harmonic. Nodes are located at every 0.80/n meters, where n is the harmonic number, while antinodes are located at every 0.80 \cdot (m + 1/2) / n meters, where m and n are integers.

Step-by-step solution

Calculate the frequency of the fundamental and third harmonic

The frequency of a vibrating string is given by \(f = v/2L\), where \(v\) is the wave speed and \(L\) is the length of the string. Using the provided wave speed (\(v = 100\) m/s) and string length (\(L = 0.80\) m), calculate the frequency of the fundamental mode (first frequency), \(f_1\). Also calculate the frequency of the third harmonic, \(f_3\), which is three times the first frequency.

Write the equation of the fundamental and third harmonic

The equation of a vibrating string in its nth mode is given by \(y(x, t) = A \sin(nx \pi / L) \cos(nft \pi)\), where \(A\) is the amplitude, \(n\) is the harmonic number, \(x\) is the position along the string, and \(t\) is time. Write down the equation for the fundamental mode (\(n=1\)) and the third harmonic (\(n=3\)), using the calculated frequencies and the provided amplitudes.

Find the nodes and antinodes

Nodes occur where the string does not move, i.e., where \(y(x, t) = 0\). This happens when \(x = L \cdot m / n\), where \(m\) is an integer. For both the fundamental mode and the third harmonic, figure out where their nodes are. Antinodes occur where the string's movement is maximum, i.e., where \(|y(x, t)|\) is at its peak. This happens when \(x = L \cdot (m + 1/2) / n\). Calculate the location of the antinodes for both the fundamental mode and the third harmonic.

Key concepts

Fundamental Frequency Calculation

The fundamental frequency of a vibrating string is essential to understanding the behavior of stringed musical instruments, such as guitars and violins. It represents the lowest frequency at which the string vibrates, producing the deepest tone that the string can offer. To calculate the fundamental frequency (\f_1\f), use the formula:
$$ f_1 = \frac{v}{2L} $$
where

• \f_1\frac is the fundamental frequency,
• \f_v\f is the wave speed, and
• \f_L\f is the length of the string.

In our exercise, the wave speed is given as 100 m/s, and the length of the string as 80 cm (which is 0.80 m). Plugging in the values, the fundamental frequency is calculated as follows:
$$ f_1 = \frac{100 \text{ m/s}}{2 \times 0.80 \text{ m}} = \frac{100}{1.6} \text{ Hz} = 62.5 \text{ Hz} $$
This is the frequency you would hear if you plucked the string and let it vibrate freely; it's the pitch of the note that forms the basis of harmonic series on the string.

Harmonic Frequency Calculation

Harmonics are the higher frequencies at which a string can naturally vibrate, and they are multiples of the fundamental frequency, shaping the richness of the sound of the string. The harmonic frequency (\f_n\f) can be calculated using the following formula:
$$ f_n = n \times f_1 $$
where

• \f_n\f is the nth harmonic frequency,
• \f_n\f is the number of the harmonic, and
• \f_f_1\f is the fundamental frequency.

In our problem, we're calculating the third harmonic frequency (\f_f_3\frac), which is three times the fundamental frequency. Using the fundamental frequency we already calculated (62.5 Hz), the third harmonic (\f_f_3\f) is found to be:
$$ f_3 = 3 \times f_1 = 3 \times 62.5 \text{ Hz} = 187.5 \text{ Hz} $$
This gives us the frequency for the third harmonic, which is higher in pitch and contributes to the overtone series that defines the unique tone of the string when played.

Wave Equation for a String

The wave equation for a string provides us with a mathematical model to describe the motion of the string as it vibrates. For a string vibrating in its nth mode or harmonic, the wave equation is represented as:
$$ y(x, t) = A \times \text{sin}\bigg(\frac{n x \times \f\frac}{L}\bigg) \times \text{cos}\bigg(n \times f \times t \times \f\frac \frac\bigg) $$
Here,

• \f_A\f is the amplitude of the wave,
• \f_x\f is the position along the string from one end,
• \f_t\f is the time,
• \f_n\f is the harmonic number (1 for the fundamental frequency),
• \f_f\f is the frequency,
• \f_\frac\f is the symbol for pi, which is approximately 3.14159.

Using the calculated fundamental and third harmonic frequencies, along with the given amplitudes, the equations for our string's fundamental frequency and the third harmonic are expressed as:
$$ y(x, t)_{f_1} = 2 \text{ cm} \times \text{sin}\bigg(\frac{x \times \f\frac}{0.80 \text{ m}}\bigg) \times \text{cos}\bigg(62.5 \text{ Hz} \times t \times \f\frac \frac\bigg) $$
and
$$ y(x, t)_{f_3} = 1 \text{ cm} \times \text{sin}\bigg(\frac{3 x \times \f\frac}{0.80 \text{ m}}\bigg) \times \text{cos}\bigg(187.5 \text{ Hz} \times t \times \f\frac \frac\bigg) $$
This accurately models the position of any point on the string at any given instant, reflecting the harmonious dance of a string's vibration.

Nodes and Antinodes of a Vibrating String

In the study of a vibrating string's harmonics, nodes and antinodes play a crucial role. Nodes are points where the string has zero amplitude and thus remains stationary during vibration. Antinodes, on the other hand, are points where the amplitude reaches a maximum, resulting in the largest motion. For a string vibrating at its nth harmonic, nodes are evenly spaced along its length, located at positions calculated with:
$$ x_{node} = \frac{L \times m}{n} $$
where

• \f_m\f is an integer representing the sequential number of the node along the string (starting from 0),
• \f_L\f is the length of the string, and
• \f_n\f is the harmonic number.

Antinodes, located between nodes, can be found at points given by:
$$ x_{antinode} = \frac{L \times (m + \frac{1}{2})}{n} $$
For the fundamental mode (\f_n=1\f) and the third harmonic (\f_n=3\f), as in our exercise,

• The nodes of the fundamental mode are at every point where \f_m\f is a whole number.
• The antinodes are halfway between these nodes.
• For the third harmonic, there are three times as many nodes and antinodes within the same string length.

Using these equations, we can locate the exact points of the nodes and antinodes for both modes of vibration, providing a visual representation of the string's harmonic behavior.