Download original notes
PAGE 1

Wave Eq. (part 2)

\[ y_{tt} = a^2 y_{xx} \quad 0 < x < L \quad t > 0 \]

\( y(0,t) = y(L,t) = 0 \) ends fixed

\( y(x,0) = f(x) \) initial displacement

\( y_t(x,0) = g(x) \) initial velocity

A diagram of a vibrating string fixed at endpoints x equals 0 and x equals L. A curved line represents the displacement y of x comma t above the horizontal axis y equals 0.

last time: \( g(x) = 0 \), displacement only ("Problem A")

\[ y(x,t) = \sum_{n=1}^{\infty} A_n \cos\left(\frac{n\pi a}{L}t\right) \sin\left(\frac{n\pi}{L}x\right) \]
\[ A_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi}{L}x\right) dx \]

today: \( f(x) = 0 \) initial velocity only ("Problem B")

PAGE 2

Same basic idea: \( y(x,t) = X(x)T(t) \)

⋮
\[ \frac{X''}{X} = \frac{T''}{a^2 T} = -\lambda \]

two ODEs:

\[ X'' + \lambda X = 0 \] \[ T'' + a^2 \lambda T = 0 \]

same BCs:

\( y(0,t) = 0 \rightarrow X(0)T(t) = 0 \rightarrow X(0) = 0 \)

\( y(L,t) = 0 \rightarrow X(L)T(t) = 0 \rightarrow X(L) = 0 \)

same spatial solution:

\[ \lambda_n = \frac{n^2 \pi^2}{L^2} \quad n = 1, 2, 3, \dots \]

\[ X_n = \sin\left(\frac{n\pi}{L}x\right) \]

\[ T'' + \frac{a^2 n^2 \pi^2}{L^2} T = 0 \]
PAGE 3

IC: no initial displacement \(\rightarrow y(x,0) = 0\)

\(X(x)T(0) = 0 \rightarrow T(0) = 0\)

\[ T(t) = A \cos\left(\frac{n\pi a}{L}t\right) + B \sin\left(\frac{n\pi a}{L}t\right) \]

w/ \(T(0) = 0 \rightarrow A = 0\)

so, \(T_n = \sin\left(\frac{n\pi a}{L}t\right)\) (Problem A we had \(T_n = \cos\left(\frac{n\pi a}{L}t\right)\))

for each \(n\), \(y_n = \sin\left(\frac{n\pi a}{L}t\right) \sin\left(\frac{n\pi}{L}x\right)\)

General Solution:

\[ y(x,t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi a}{L}t\right) \sin\left(\frac{n\pi}{L}x\right) \]

last IC: \(y_t(x,0) = g(x)\)

\[ y_t(x,t) = \sum_{n=1}^{\infty} \frac{n\pi a}{L} B_n \cos\left(\frac{n\pi a}{L}t\right) \sin\left(\frac{n\pi}{L}x\right) \]

\[ g(x) = \sum_{n=1}^{\infty} \left( \frac{n\pi a}{L} B_n \right) \sin\left(\frac{n\pi}{L}x\right) \]
sine series coeff: \(\frac{n\pi a}{L} B_n\)

\[ \frac{n\pi a}{L} B_n = \frac{2}{L} \int_{0}^{L} g(x) \sin\left(\frac{n\pi}{L}x\right) dx \]

PAGE 4
\[ B_n = \frac{2}{n\pi a} \int_{0}^{L} g(x) \sin\left(\frac{n\pi}{L}x\right) dx \]

Example

A string w/ \(L=1\), \(a=1\) is initially at rest.

It is struck w/ a hammer width 0.2 at \(x=0.5\) (center of string) w/ upward velocity of 10.

A diagram of a string fixed at x=0 and x=1. An upward arrow at the midpoint x=0.5 represents the strike, with a box below showing the hammer width of 0.2.

initial velocity \(g(x) = \begin{cases} 10 & 0.4 < x < 0.6 \\ 0 & \text{else} \end{cases}\)

A coordinate graph of g(x) versus x. A rectangular pulse is shown with a height of 10, starting at x=0.4 and ending at x=0.6.

\(\vdots\)

\[ y(x,t) = \sum_{n=1}^{\infty} \frac{20}{n^2\pi^2} \left[ \cos(0.4n\pi) - \cos(0.6n\pi) \right] \sin(n\pi t) \sin(n\pi x) \]
\(\uparrow\)
freq. of each mode (\(n\))
PAGE 5

String Displacement \(y(x, t)\)

The following 3D surface plot illustrates the displacement of a string, denoted as \(y(x, t)\), as a function of both position \(x\) and time \(t\). The visualization shows how a wave pulse propagates and changes shape over the spatial domain \(x \in [0, 1]\) and temporal domain \(t \in [0, 2]\).

A 3D surface plot showing string displacement y as a function of position x and time t. The surface starts as a narrow peak at early times and widens into a trapezoidal pulse that moves across the position axis as time increases. The vertical axis represents displacement from -1.00 to 1.00, the horizontal axis represents position from 0.0 to 1.0, and the depth axis represents time from 0.00 to 2.00.
PAGE 6

Snapshots of String Displacement \(y(x)\) at different times

This 2D plot provides a series of snapshots showing the string's displacement \(y(x)\) at specific time intervals. It highlights the evolution of the wave profile from a triangular peak to a broader trapezoidal shape as it moves through the medium.

A 2D line graph showing multiple snapshots of string displacement y versus position x. Different colored lines represent different times: t=0.1 (blue peak), t=0.2 (orange trapezoid), t=0.3 (yellow trapezoid), t=0.5 (purple trapezoid), and t=0.7 (brown trapezoid). The displacement y ranges from 0.0 to 1.0, and position x ranges from 0.0 to 1.0.

Time Snapshots Legend:

  • \(t = 0.1\)
  • \(t = 0.2\)
  • \(t = 0.3\)
  • \(t = 0.5\)
  • \(t = 0.7\)
PAGE 7

Motion of Specific String Points \(y(t)\)

The following graph illustrates the displacement \(y\) over time \(t\) for three specific points along a vibrating string, located at positions \(x = 0.2\), \(x = 0.5\), and \(x = 0.8\).

A line graph showing displacement y versus time t for three points on a string. The x-axis ranges from 0.0 to 4.0, and the y-axis ranges from -1.00 to 1.00. Three periodic, trapezoidal-like waveforms are shown for x=0.2, x=0.5, and x=0.8, demonstrating phase shifts and varying durations of peak displacement.
PAGE 8

String Displacement at \(t = 0.2\): Total Solution vs. Harmonics

This visualization compares the total solution for string displacement (calculated with \(n=100\) terms) against individual odd harmonics (\(n=1\), \(n=3\), \(n=5\), and \(n=7\)) at a specific snapshot in time, \(t = 0.2\).

A line graph showing displacement y versus position x at time t=0.2. A solid black line represents the total solution (n=100), showing a trapezoidal pulse. Several dashed colored lines represent individual harmonics: n=1 (blue), n=3 (red), n=5 (green), and n=7 (yellow), illustrating how the summation of sine waves approximates the complex pulse shape.
PAGE 9

Physical Interpretation of Harmonics

the solution we got has no even harmonics (even \(n\)'s)

why (physically)?

\(n = 1\)
A standing wave diagram for the first harmonic (n=1) showing a single large loop between two fixed endpoints. A dot marks the center of the string at its maximum displacement.
\(n = 2\)
A standing wave diagram for the second harmonic (n=2) showing two loops with a node in the center. The center point of the string is fixed and does not move.
\(n = 3\)
A standing wave diagram for the third harmonic (n=3) showing three loops. The center of the string is at a point of maximum displacement (antinode).
\(n = 4\)
A standing wave diagram for the fourth harmonic (n=4) showing four loops with a node in the center.

we hit the string at \(x = 0.5\)

\(\rightarrow\) puts that point in motion

notice for \(n = \text{even}\), the center cannot move

hitting center = destroying all even modes

(same if displaced)

on a real piano the impact point is \(\frac{L}{7}\) or \(\frac{L}{9}\)

\(\downarrow\)

eliminates all multiples of 7th mode because 7th and 9th sound dissonant

PAGE 10

Problem A: displacement only

Problem B: velocity only

General Case

simply add up the solution

because \(y_{tt} = a^2 y_{xx}\) is linear

sound from wind instruments \(\rightarrow\) pressure waves

eg:
\[ \frac{\partial^2 p}{\partial t^2} = a^2 \frac{\partial^2 p}{\partial x^2} \]

\(p(x,t) = \text{pressure}\)

same equation!