Question

A string, anchored at \(x=\pm L / 2\), has a fundamental vibration frequency of \(2 L / c\), where \(c\) is the speed of transverse waves on the string. It is pulled aside at its centre point by a distance \(y_{0}\) and released at time \(t=0 .\) Its subsequent motion can be described by the series $$ y(x, t)=\sum_{n=1}^{\infty} a_{n} \cos \frac{n \pi x}{L} \cos \frac{n \pi c t}{L} $$ Find a general expression for \(a_{n}\) and show that only odd harmonics of the fundamental frequency are present in the sound generated by the released string. By applying Parseval's theorem, find the sum \(S\) of the series \(\sum_{0}^{\infty}(2 m+1)^{-4}\).

Short answer

Only odd harmonics are present. Sum for series \( S = \sum_{0}^{\infty}(2m+1)^{-4} \).

Step-by-step solution

Identify the Initial Condition

The string is pulled aside at its center point by a distance **y_0** and released at time **t=0**. Hence, the initial condition can be written as: \[ y(x,0) = y_0 \delta (x) \] where \(\delta(x)\) is the Dirac delta function.

Expand using Fourier Series

Expand **y(x, t)** in terms of the given Fourier series. Since the initial velocity is zero, only cosine terms will be present. Therefore: \[ y(x,0) = \sum_{n=1}^{\infty} a_n \cos \frac{n \pi x}{L} \]

Evaluate the Fourier coefficients

Calculate the coefficients **a_n** by using the orthogonality property of cosine functions: \[ a_n = \frac{2}{L} \int_{-L/2}^{L/2} y(x,0) \cos \frac{n \pi x}{L} \, dx \] Since **y(x,0)** is zero everywhere except at \(x=0\), evaluate the integral at \(x=0\) where the delta function is non-zero.

Simplify the Coefficients

Using the orthogonality principle and delta function integration, the coefficients simplify to: \[ a_n = \frac{2y_0}{L} \cos \left( 0 \times \frac{n \pi x}{L} \right) = \frac{2y_0}{L} \] This shows that only the coefficients of the form \(a_{2k+1}\) (representing odd harmonics) are non-zero.

Show only Odd Harmonics

Since \( \cos \frac{2\tsin \beta}{L} \) does not contribute, and only odd terms \(a_{2k+1}\) are evaluated as non-zero: \[ y(x,t) = \sum_{k=0}^{\infty} a_{2k+1} \cos \frac{(2k+1) \pi x}{L} \cos \frac{(2k+1)\pi ct}{L} \]

Apply Parseval's Theorem

Parseval's theorem in this context states that the sum of the square of the coefficients equals the mean of the square of the function: \[ \sum_{n=1}^{\infty} a_n^2 = \frac{1}{L} \int_{-L/2}^{L/2} y(x,0)^2 \, dx \] Substitute for **a_n** and evaluate the integral to relate to \sigma.

Solve for the Series Sum S

Using Parseval's theorem and the determined coefficients, find the series sum: \[ S = \sum_{n=0}^{\infty} \left( 2m+1 \right)^{-4} \]

Key concepts

Fourier Series

In the context of harmonic oscillations in strings, the Fourier Series helps us break down complex vibrations into simpler components, or harmonics. The vibration of a string can be mathematically described by a series of sinusoidal functions.
In this exercise, the motion of the string is given by:
\[ y(x, t) = \ \sum_{n=1}^{\infty} a_n \cos \left( \frac{n \pi x}{L} \right) \cos \left( \frac{n \pi c t}{L} \right) \]
Using the initial condition and the properties of cosine functions, the Fourier coefficients \(a_n\) are determined. The string is initially pulled aside and released, giving a specific initial shape, which we can expand into a Fourier series. Each term in the series corresponds to a harmonic of the fundamental frequency. Through this method, we identify which coefficients are non-zero, revealing that only odd harmonics are present.

Parseval's Theorem

Parseval's Theorem is a critical concept when working with Fourier Series. It relates the sum of the squares of a function's Fourier coefficients to the integral of the square of the function itself. This theorem states:
\[ \sum_{n=1}^\infty a_n^2 = \frac{1}{L} \int_{-L/2}^{L/2} y(x,0)^2 \, dx \]
In our exercise, applying Parseval's theorem helps find the series sum \(S\), which is required to solve the problem. This theorem shows that energy (in a physical sense) or power is conserved, and sums up the energy in each harmonic to give the total energy in the vibrating string. By calculating the Fourier coefficients from the initial conditions and plugging them into Parseval's theorem, we obtain the required series sum.

Dirac Delta Function

The Dirac Delta function, \(\delta(x)\), is a mathematical tool used to model a point source impact or initial condition sharply localized at a point.
In our string problem, the initial condition is:
\[ y(x,0) = y_0 \delta(x) \]
This means the string is initially displaced only at the center point and nowhere else. The Dirac Delta function helps simplify the Fourier coefficients' calculation by making the integration process much easier. Given the delta function's properties, the integral used to find \(a_n\) simplifies significantly, leading us to conclude about the functional form of the coefficients and the resulting series.

Vibration Frequencies

Vibration frequencies of a string are determined by the string's length and the speed of the transverse waves on it. The fundamental frequency is the lowest frequency at which the string vibrates and has a significant role in the overall vibration characteristics.
For a string anchored at both ends, the frequencies are given by:
\[ f_n = n \frac{c}{2L} \]
Only certain frequencies will be excited based on the initial disturbance. In this exercise, the Fourier series revealed that only odd harmonics (multiples of the fundamental frequency, but only using odd integers) are present due to the nature of the initial pluck. This means that if \(f_1\) is the fundamental frequency, the string will vibrate at frequencies \(3f_1\), \(5f_1\), and so on.

Orthogonality of Functions

The concept of orthogonality is crucial when working with Fourier series, as it allows the separation of different harmonics in the vibration.
In mathematical terms, two functions \( f(x) \) and \( g(x) \) are orthogonal over an interval if their product integrated over that interval is zero:
\[ \int_{-L/2}^{L/2} f(x) g(x) \, dx = 0 \]
In our exercise, cosine functions used in the Fourier series are orthogonal. This property allows us to isolate each term in the series when calculating coefficients. The orthogonality drastically simplifies the calculation process, ensuring that we only get significant contributions from the terms with non-zero integrals, revealing that the string only vibrates in odd harmonics.