Question

Consider the representation as a Fourier series of the displacement of a string lying in the interval \(0 \leq x \leq L\) and fixed at its ends, when it is pulled aside by \(y_{0}\) at the point \(x=L / 4\). Sketch the continuations for the region outside the interval that will (a) produce a series of period \(L\), (b) produce a series that is antisymmetric about \(x=0\), and (c) produce a series that will contain only cosine terms. (d) What are (i) the periods of the series in (b) and (c) and (ii) the value of the \({ }^{4} a_{0}\)-term' in (c)? (e) Show that a typical term of the series obtained in (b) is $$ \frac{32 y_{0}}{3 n^{2} \pi^{2}} \sin \frac{n \pi}{4} \sin \frac{n \pi x}{L} $$

Short answer

Series period in (b) and (c) is \(2L\). \( a_0 \) term in (c) is 0.

Step-by-step solution

- Identify Boundary Conditions

The string is fixed at both ends, so we have boundary conditions at \( x = 0 \) and \( x = L \): \( y(0) = 0 \) and \( y(L) = 0 \).

- Sketch Continuation for Period L

To find the continuation that produces a series of period \( L \), we repeat the function in intervals of width \( L \).

- Sketch Continuation for Antisymmetric Series

For a series antisymmetric around \( x = 0 \), reflect the function across the y-axis and reverse its sign. This creates a full period of \( 2L \).

- Sketch Continuation for Cosine Terms Only

To have only cosine terms, reflect the function about the vertical line at \( x = L/2 \), making it an even function. This will produce a period of \( 2L \).

- Find Periods of Series

(i) The period for the antisymmetric series (b) is \( 2L \). (ii) The period for the series containing only cosine terms (c) is also \( 2L \).

- Compute Value of \( a_0 \)-term in (c)

Since \( a_0 \) represents the average value of the function over one period, which is zero due to symmetry, \( a_0 = 0 \).

- Verify Typical Term for Antisymmetric Series

Given the conditions in (b), the Fourier series has the general form of sine terms. Confirm that \[ \frac{32 y_0}{3 n^2 \pi^2} \sin \frac{n \pi}{4} \sin \frac{n \pi x}{L} \] fits as a typical term by evaluating its coefficients and checking the boundary conditions.

Key concepts

Boundary Conditions

To understand the Fourier series representation of the string's displacement, we must first consider boundary conditions. Boundary conditions are the constraints that are applied to the endpoints of a system, such as a vibrating string. For a string fixed at both ends, the displacement must be zero at the endpoints. This means that at positions \(x = 0\) and \(x = L\), the displacement \(y(x)\) must satisfy:
\[ y(0) = 0 \]
\[ y(L) = 0 \]
This controls the possible shapes that the string can take and determines the form of the Fourier series.

Periodic Functions

A periodic function consistently repeats its values in regular intervals. For our string, to find the continuation that produces a series of period \(L\), we repeat the function at intervals of width \(L\). Imagine duplicating the initial displacement shape of the string over and over along the x-axis. This repetition mimics the wave nature and helps in simplifying complex vibrations into sums of simpler periodic waves.
The Fourier series is particularly powerful for periodic functions because it allows us to express these functions as sums of sines and cosines with periods that divide the period of the function.

Antisymmetric Functions

Antisymmetric functions are those that satisfy the condition: \( f(x) = -f(-x) \). In other words, the function is symmetric about the origin but with inverted signs. For our string, the antisymmetric series is created by reflecting the function across the y-axis and reversing its sign. This creates a full period of \(2L\).
For example, if we had a displacement at \( x = \frac{L}{4}\), its antisymmetric counterpart might look like this at \( x = - \frac{L}{4}\) producing a form where positive displacements on one side of the origin are matched with negative displacements on the other side.

Cosine Terms

To create a Fourier series containing only cosine terms, the function must be even. This means it must satisfy: \( f(x) = f(-x) \). For our string, we achieve this by reflecting the function about the vertical line at \(x = \frac{L}{2}\), making it even. This operation keeps the displacements symmetric relative to the midpoint. This transformation makes the function periodic with period \(2L\).
Cosine terms represent symmetrical oscillations around the midpoint, helping us focus on those vibration modes that repeat every half period.

Fourier Coefficients

Fourier coefficients quantify the contribution of each sinusoidal component to the overall shape of the function. For the string's displacement, the coefficients \(a_n\) and \(b_n\) are calculated from the original function. They provide weights to the corresponding cosine and sine terms in the Fourier series.
Specifically:

• \(a_n\) coefficients correspond to the cosine terms, linked to even functions.
• \(b_n\) coefficients correspond to the sine terms, which are important for odd functions like antisymmetric cases.

Remember, these coefficients simplify and transform real physical problems into manageable mathematical forms.