Expressing the wave equation solution by separation of variables as a superposition of forward and backward waves.Solving the wave equation with Neumann conditionsWhat condition does this impose on the coefficients $ B_n$ in the vibrating string problemWhen solving the wave equation by separation of variables, is the separation constant always negative?Method of separation of variables in a wave problemFinding particular solution to 1D - wave equation given general solution.Wave Equation by separation of variablesSolving the wave equation with Neumann boundary conditionsRigorous solution of wave equation for string with fixed endsFinding the constants $A$ and $B$ for the general series solution of the wave equation with initial condition $g(x) = max 8x − 4$A partial solution to the wave equation
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Expressing the wave equation solution by separation of variables as a superposition of forward and backward waves.
Solving the wave equation with Neumann conditionsWhat condition does this impose on the coefficients $ B_n$ in the vibrating string problemWhen solving the wave equation by separation of variables, is the separation constant always negative?Method of separation of variables in a wave problemFinding particular solution to 1D - wave equation given general solution.Wave Equation by separation of variablesSolving the wave equation with Neumann boundary conditionsRigorous solution of wave equation for string with fixed endsFinding the constants $A$ and $B$ for the general series solution of the wave equation with initial condition $g(x) = max 8x − 4$A partial solution to the wave equation
$begingroup$
(From an exercise in Pinchover's Introduction to Partial Differential Equations).
$$u(x,t)=fracA_0 + B_0 t2+sum_n=1^infty left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL$$
is a general (and formal, at least) solution to the vibrating string with fixed ends. How to write this as a superposition of a forward and a backward wave? That is, as $f(x+ct)+f(x-ct)$ for some $f$. (No need to worry about rigour here, an heuristic will do.)
I know, by elementary trigonometry, that $$left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL =\= (1/2)(A_ncos +B_nsin)left(fraccpi ntL + fracnpi xLright)+(1/2)(A_ncos +B_nsin)left(fraccpi ntL - fracnpi xLright), $$ but this doesn't seem to work because the variable $x$ is the one that changes sign, so apparently this cannot be interpreted as a sum of forward and backward waves.
Is there a workaround to this?
EDIT. The second wave is from another function $g$. The answer is then straightforward after oen's comment.
analysis pde physics fourier-series wave-equation
edited yesterday
Andrews
1,2641420
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
$endgroup$
|
show 1 more comment
$begingroup$
(From an exercise in Pinchover's Introduction to Partial Differential Equations).
$$u(x,t)=fracA_0 + B_0 t2+sum_n=1^infty left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL$$
is a general (and formal, at least) solution to the vibrating string with fixed ends. How to write this as a superposition of a forward and a backward wave? That is, as $f(x+ct)+f(x-ct)$ for some $f$. (No need to worry about rigour here, an heuristic will do.)
I know, by elementary trigonometry, that $$left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL =\= (1/2)(A_ncos +B_nsin)left(fraccpi ntL + fracnpi xLright)+(1/2)(A_ncos +B_nsin)left(fraccpi ntL - fracnpi xLright), $$ but this doesn't seem to work because the variable $x$ is the one that changes sign, so apparently this cannot be interpreted as a sum of forward and backward waves.
Is there a workaround to this?
EDIT. The second wave is from another function $g$. The answer is then straightforward after oen's comment.
analysis pde physics fourier-series wave-equation
edited yesterday
Andrews
1,2641420
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
$endgroup$
2
$begingroup$
Wait a minute: shouldn't the superposition of arbitrary forward and backward waves be of the form $f(x+ct)+g(x-ct)$? Then oen's observation should help you resolve the issue...
$endgroup$
– anon
Jul 17 '12 at 1:29
$begingroup$
@anon You're right. Stupid me.
$endgroup$
– Weltschmerz
Jul 17 '12 at 1:42
$begingroup$
@anon: Thanks for the careful reading. I skipped right over $f(x+t) + f(x-t)$ ...
$endgroup$
– user26872
Jul 17 '12 at 1:44
$begingroup$
@Weltschmerz: Typos happen to everybody!
$endgroup$
– user26872
Jul 17 '12 at 1:46
2
$begingroup$
Write $$a+bt=v(x+ct)+w(x-ct)=(v+w)x+(v-w)ct$$ where $v,w$ are constants and solve the subsequent linear system.
$endgroup$
– anon
Jul 17 '12 at 2:23
|
show 1 more comment
$begingroup$
(From an exercise in Pinchover's Introduction to Partial Differential Equations).
$$u(x,t)=fracA_0 + B_0 t2+sum_n=1^infty left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL$$
is a general (and formal, at least) solution to the vibrating string with fixed ends. How to write this as a superposition of a forward and a backward wave? That is, as $f(x+ct)+f(x-ct)$ for some $f$. (No need to worry about rigour here, an heuristic will do.)
I know, by elementary trigonometry, that $$left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL =\= (1/2)(A_ncos +B_nsin)left(fraccpi ntL + fracnpi xLright)+(1/2)(A_ncos +B_nsin)left(fraccpi ntL - fracnpi xLright), $$ but this doesn't seem to work because the variable $x$ is the one that changes sign, so apparently this cannot be interpreted as a sum of forward and backward waves.
Is there a workaround to this?
EDIT. The second wave is from another function $g$. The answer is then straightforward after oen's comment.
analysis pde physics fourier-series wave-equation
edited yesterday
Andrews
1,2641420
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
$endgroup$
(From an exercise in Pinchover's Introduction to Partial Differential Equations).
$$u(x,t)=fracA_0 + B_0 t2+sum_n=1^infty left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL$$
is a general (and formal, at least) solution to the vibrating string with fixed ends. How to write this as a superposition of a forward and a backward wave? That is, as $f(x+ct)+f(x-ct)$ for some $f$. (No need to worry about rigour here, an heuristic will do.)
I know, by elementary trigonometry, that $$left(A_ncosfraccpi ntL+ B_nsinfraccpi ntLright)cosfracnpi xL =\= (1/2)(A_ncos +B_nsin)left(fraccpi ntL + fracnpi xLright)+(1/2)(A_ncos +B_nsin)left(fraccpi ntL - fracnpi xLright), $$ but this doesn't seem to work because the variable $x$ is the one that changes sign, so apparently this cannot be interpreted as a sum of forward and backward waves.
Is there a workaround to this?
EDIT. The second wave is from another function $g$. The answer is then straightforward after oen's comment.
analysis pde physics fourier-series wave-equation
analysis pde physics fourier-series wave-equation
edited yesterday
Andrews
1,2641420
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
edited yesterday
Andrews
1,2641420
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
edited yesterday
Andrews
1,2641420
edited yesterday
Andrews
1,2641420
edited yesterday
Andrews
1,2641420
1,2641420
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
asked Jul 17 '12 at 1:14
WeltschmerzWeltschmerz
4,0622041
4,0622041
2
$begingroup$
Wait a minute: shouldn't the superposition of arbitrary forward and backward waves be of the form $f(x+ct)+g(x-ct)$? Then oen's observation should help you resolve the issue...
$endgroup$
– anon
Jul 17 '12 at 1:29
$begingroup$
@anon You're right. Stupid me.
$endgroup$
– Weltschmerz
Jul 17 '12 at 1:42
$begingroup$
@anon: Thanks for the careful reading. I skipped right over $f(x+t) + f(x-t)$ ...
$endgroup$
– user26872
Jul 17 '12 at 1:44
$begingroup$
@Weltschmerz: Typos happen to everybody!
$endgroup$
– user26872
Jul 17 '12 at 1:46
2
$begingroup$
Write $$a+bt=v(x+ct)+w(x-ct)=(v+w)x+(v-w)ct$$ where $v,w$ are constants and solve the subsequent linear system.
$endgroup$
– anon
Jul 17 '12 at 2:23
|
show 1 more comment
2
$begingroup$
Wait a minute: shouldn't the superposition of arbitrary forward and backward waves be of the form $f(x+ct)+g(x-ct)$? Then oen's observation should help you resolve the issue...
$endgroup$
– anon
Jul 17 '12 at 1:29
$begingroup$
@anon You're right. Stupid me.
$endgroup$
– Weltschmerz
Jul 17 '12 at 1:42
$begingroup$
@anon: Thanks for the careful reading. I skipped right over $f(x+t) + f(x-t)$ ...
$endgroup$
– user26872
Jul 17 '12 at 1:44
$begingroup$
@Weltschmerz: Typos happen to everybody!
$endgroup$
– user26872
Jul 17 '12 at 1:46
2
$begingroup$
Write $$a+bt=v(x+ct)+w(x-ct)=(v+w)x+(v-w)ct$$ where $v,w$ are constants and solve the subsequent linear system.
$endgroup$
– anon
Jul 17 '12 at 2:23
2
2
$begingroup$
Wait a minute: shouldn't the superposition of arbitrary forward and backward waves be of the form $f(x+ct)+g(x-ct)$? Then oen's observation should help you resolve the issue...
$endgroup$
– anon
Jul 17 '12 at 1:29
$begingroup$
Wait a minute: shouldn't the superposition of arbitrary forward and backward waves be of the form $f(x+ct)+g(x-ct)$? Then oen's observation should help you resolve the issue...
$endgroup$
– anon
Jul 17 '12 at 1:29
$begingroup$
@anon You're right. Stupid me.
$endgroup$
– Weltschmerz
Jul 17 '12 at 1:42
$begingroup$
@anon You're right. Stupid me.
$endgroup$
– Weltschmerz
Jul 17 '12 at 1:42
$begingroup$
@anon: Thanks for the careful reading. I skipped right over $f(x+t) + f(x-t)$ ...
$endgroup$
– user26872
Jul 17 '12 at 1:44
$begingroup$
@anon: Thanks for the careful reading. I skipped right over $f(x+t) + f(x-t)$ ...
$endgroup$
– user26872
Jul 17 '12 at 1:44
$begingroup$
@Weltschmerz: Typos happen to everybody!
$endgroup$
– user26872
Jul 17 '12 at 1:46
$begingroup$
@Weltschmerz: Typos happen to everybody!
$endgroup$
– user26872
Jul 17 '12 at 1:46
2
2
$begingroup$
Write $$a+bt=v(x+ct)+w(x-ct)=(v+w)x+(v-w)ct$$ where $v,w$ are constants and solve the subsequent linear system.
$endgroup$
– anon
Jul 17 '12 at 2:23
$begingroup$
Write $$a+bt=v(x+ct)+w(x-ct)=(v+w)x+(v-w)ct$$ where $v,w$ are constants and solve the subsequent linear system.
$endgroup$
– anon
Jul 17 '12 at 2:23
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Notice that $cos(t-x) = cos(x-t)$ and $sin(t-x) = -sin(x-t)$.
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
$endgroup$
add a comment |
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$begingroup$
Notice that $cos(t-x) = cos(x-t)$ and $sin(t-x) = -sin(x-t)$.
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
$endgroup$
add a comment |
$begingroup$
Notice that $cos(t-x) = cos(x-t)$ and $sin(t-x) = -sin(x-t)$.
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
$endgroup$
add a comment |
$begingroup$
Notice that $cos(t-x) = cos(x-t)$ and $sin(t-x) = -sin(x-t)$.
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
$endgroup$
Notice that $cos(t-x) = cos(x-t)$ and $sin(t-x) = -sin(x-t)$.
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
answered Jul 17 '12 at 1:20
user26872user26872
14.9k22773
14.9k22773
add a comment |
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2
$begingroup$
Wait a minute: shouldn't the superposition of arbitrary forward and backward waves be of the form $f(x+ct)+g(x-ct)$? Then oen's observation should help you resolve the issue...
$endgroup$
– anon
Jul 17 '12 at 1:29
$begingroup$
@anon You're right. Stupid me.
$endgroup$
– Weltschmerz
Jul 17 '12 at 1:42
$begingroup$
@anon: Thanks for the careful reading. I skipped right over $f(x+t) + f(x-t)$ ...
$endgroup$
– user26872
Jul 17 '12 at 1:44
$begingroup$
@Weltschmerz: Typos happen to everybody!
$endgroup$
– user26872
Jul 17 '12 at 1:46
2
$begingroup$
Write $$a+bt=v(x+ct)+w(x-ct)=(v+w)x+(v-w)ct$$ where $v,w$ are constants and solve the subsequent linear system.
$endgroup$
– anon
Jul 17 '12 at 2:23