5. Suppose that we have an infinite string, and that we know the initial position and velocity of the string for all the points x € [−1,1] (and we do not have any information on f and g outside this interval). (a) For what portion of the string can we find u(x, t) at time t = 1, if the speed of wave propagation is c = 1/2? (b) Up to what time can we say anything about any points along the string? (In other words, what is the largest value, t₁, for which we know u(x₁, t₁) for at least one point x₁?) Hint: Find the point, (x₁, t₁) whose domain of dependence for t = 0 is [−1, 1].
5. Suppose that we have an infinite string, and that we know the initial position and velocity of the string for all the points x € [−1,1] (and we do not have any information on f and g outside this interval). (a) For what portion of the string can we find u(x, t) at time t = 1, if the speed of wave propagation is c = 1/2? (b) Up to what time can we say anything about any points along the string? (In other words, what is the largest value, t₁, for which we know u(x₁, t₁) for at least one point x₁?) Hint: Find the point, (x₁, t₁) whose domain of dependence for t = 0 is [−1, 1].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Suppose that we have an infinite string, and that we know the initial
position and velocity of the string for all the points x € [−1, 1] (and we do
not have any information on f and g outside this interval). (a) For what
portion of the string can we find u(x, t) at time t 1, if the speed of wave
propagation is c = 1/2? (b) Up to what time can we say anything about
any points along the string? (In other words, what is the largest value, t₁,
for which we know u(x₁, t₁) for at least one point x₁?) Hint: Find the point,
(x1, t₁) whose domain of dependence for t 0 is [1,1].
-
Solution: We can draw a triangle with vertices at points (−1,0), (1,0),
and (0,2) in the x t diagram. The left and righ sides are characteristics
with slopes 1/c 2. This is the domain of dependence of point (0, 2). For
the point (0, 2), the domain of dependence for t = 0 is the interval [−1, 1].
For any point with t > 2, the domain of dependence at t = 0 will not be
contained in [−1,1]. This means that for t > 2, we cannot find the solution
u(x, t) for any value of x (we do not have enough information for it). This
answers question (b). For (a), we draw a horizontal line at t
1. It crosses
the triangle at the points (-1/2, 1) and (1/2, 1). It is easy to check that the
domains of dependence of all the points on this line with x € [−1/2,1/2]
are contained in the large triangle. In particular, for t = 0, their domains of
dependence are inside [−1,1]. Therefore, for t = 1, we can write down the
solution for points with x € [-1/2, 1/2].
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Transcribed Image Text:=
5. Suppose that we have an infinite string, and that we know the initial
position and velocity of the string for all the points x € [−1, 1] (and we do
not have any information on f and g outside this interval). (a) For what
portion of the string can we find u(x, t) at time t 1, if the speed of wave
propagation is c = 1/2? (b) Up to what time can we say anything about
any points along the string? (In other words, what is the largest value, t₁,
for which we know u(x₁, t₁) for at least one point x₁?) Hint: Find the point,
(x1, t₁) whose domain of dependence for t 0 is [1,1].
-
Solution: We can draw a triangle with vertices at points (−1,0), (1,0),
and (0,2) in the x t diagram. The left and righ sides are characteristics
with slopes 1/c 2. This is the domain of dependence of point (0, 2). For
the point (0, 2), the domain of dependence for t = 0 is the interval [−1, 1].
For any point with t > 2, the domain of dependence at t = 0 will not be
contained in [−1,1]. This means that for t > 2, we cannot find the solution
u(x, t) for any value of x (we do not have enough information for it). This
answers question (b). For (a), we draw a horizontal line at t
1. It crosses
the triangle at the points (-1/2, 1) and (1/2, 1). It is easy to check that the
domains of dependence of all the points on this line with x € [−1/2,1/2]
are contained in the large triangle. In particular, for t = 0, their domains of
dependence are inside [−1,1]. Therefore, for t = 1, we can write down the
solution for points with x € [-1/2, 1/2].
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