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+ (3) ar2 2. Recall from lectures that the governing PDE for vibrations of a circular...
(2 points) is typed as lambda, a as alpha. The PDE a2u ar2 = yº ди...
(2 points) is typed as lambda, a as alpha. The PDE a2u ar2 = yº ди ay is separable, so we look for solutions of the form u(x, t) = X(x)Y(y). When solving DE in X and Y use the constants a and b for X and c for Y. The PDE can be rewritten using this solution as (placing constants in the DE for Y) into X"/X (1/(k^2))(y^5)(Y'/Y) = -2 Note: Use the prime notation for derivatives, so the...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacia...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a litt...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
Please follow the hint (picture 2) + 12 302 27. A drumhead is a circular membrane...
Please follow the hint (picture 2)
+ 12 302 27. A drumhead is a circular membrane of radius a. When it is struck, waves propagate across the drumhead. The membrane vibrates with displacement 5, where $(r, 0, 1) = n(r, e)e-ie and n(t,0) satisfies the Helmholtz equation 1 a an 1 aan +k?n=0 rar where k = w2/u2 and v is the speed with which waves propagate across the drumhead. (The speed v depends on the tension in the drumhead,...
I need help with d) please help thank you Question 1 Wave motion appears in all branches of physics. In the lectures we...
I need help with d) please help thank you
Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the...
12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2...
12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2 (nl) This function is of considerable importance in applied mathematics, with merous applications to problems involving cylindrical containers, such as temperature distribution in a se pipe (6) Write out the partial num for the Bessel function of order sero up to the terms. If you have access to a graphing calculator or computer, use the graph of this partial sum to approximate, to one...
Apply the method of separation of variables to the PDE below to derive a pair of...
Apply the method of separation of variables to the PDE below to derive a pair of ODEs, one of which involves only x and the other of which involves only y. (You do not need to solve the ODE.) 23 u дх3 + x 23 u dy3 = 0 6 u=o L10)=0 Cha: Supplemental information -Linearity satisfies the property Leau, uz)=C.L(ui) +C₂L(42) - Heat Egn. is a linear partial differential equation : L(a)= eu-kay = f(xt) Linear homogeneous = L()...
Problem 2. A string of a guitar is fixed at the two ends, x = 0...
Problem 2. A string of a guitar is fixed at the two ends, x = 0 and r = a. The string is set in motion with initial position f(x) = (h/a)., 0 <r <a, where h > 0, and then it is released with no initial velocity. The displacement u of the string is described by the PDE au 1 au ar2 2 212 0<x<a, t> 0. (i) State the boundary value initial value problem that u satisfies. (ii)...
please answer them in detail. The concentration C(x, t) of a pollutant in a semi-infinite pipe diffuses according to the PDE over 0<<o for > 0, where the diffusivity D is a given positive...
please answer them in detail.
The concentration C(x, t) of a pollutant in a semi-infinite pipe diffuses according to the PDE over 0<<o for > 0, where the diffusivity D is a given positive constant. An experiment is performed in which the initial concentration is zero everywhere, so that C(x, 0)0 for all r 20. Once t > 0 a fixed concentration of C(0, t) Co is imposed at x-0, where Co is a constant, and it is also assumed...
3.7 Underdamped free vibrations (e.g. a vibrating beam or slinky in damp air A system whose...
3.7 Underdamped free vibrations (e.g. a vibrating beam or slinky in damp air A system whose response is governed by the following constant-coefficient, is performing underdamped free vibrations if oc 1 (underdamped ). (Seetion 6.7). linear, 2td order, ODE ) and 1t)- (freei (a) After assuming a solution t)et where C and p are conustants, show every step that proves yit) can be expressed in terms of the yet-to-be-determined constants A and B and he constant wa (defined above) called...
(2 points) is typed as lambda, a as alpha. The PDE a2u ar2 = yº ди ay is separable, so we look for solutions of the form u(x, t) = X(x)Y(y). When solving DE in X and Y use the constants a and b for X and c for Y. The PDE can be rewritten using this solution as (placing constants in the DE for Y) into X"/X (1/(k^2))(y^5)(Y'/Y) = -2 Note: Use the prime notation for derivatives, so the...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
Please follow the hint (picture 2)
+ 12 302 27. A drumhead is a circular membrane of radius a. When it is struck, waves propagate across the drumhead. The membrane vibrates with displacement 5, where $(r, 0, 1) = n(r, e)e-ie and n(t,0) satisfies the Helmholtz equation 1 a an 1 aan +k?n=0 rar where k = w2/u2 and v is the speed with which waves propagate across the drumhead. (The speed v depends on the tension in the drumhead,...
I need help with d) please help thank you
Question 1 Wave motion appears in all branches of physics. In the lectures we considered the solution of the advection equation, a first-order hyperbolic PDE. Here we consider the solution of the wave equation: c2 where c >0 is constant. , We assume all variables have been non-dimensionalised. (a) Eq. (1) has the general solution (d'Alembert, 1747): u(x,t) F(x -ct) +G(x ct), where F and G are arbitrary functions. Consider the...
12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2 (nl) This function is of considerable importance in applied mathematics, with merous applications to problems involving cylindrical containers, such as temperature distribution in a se pipe (6) Write out the partial num for the Bessel function of order sero up to the terms. If you have access to a graphing calculator or computer, use the graph of this partial sum to approximate, to one...
Apply the method of separation of variables to the PDE below to derive a pair of ODEs, one of which involves only x and the other of which involves only y. (You do not need to solve the ODE.) 23 u дх3 + x 23 u dy3 = 0 6 u=o L10)=0 Cha: Supplemental information -Linearity satisfies the property Leau, uz)=C.L(ui) +C₂L(42) - Heat Egn. is a linear partial differential equation : L(a)= eu-kay = f(xt) Linear homogeneous = L()...
Problem 2. A string of a guitar is fixed at the two ends, x = 0 and r = a. The string is set in motion with initial position f(x) = (h/a)., 0 <r <a, where h > 0, and then it is released with no initial velocity. The displacement u of the string is described by the PDE au 1 au ar2 2 212 0<x<a, t> 0. (i) State the boundary value initial value problem that u satisfies. (ii)...
please answer them in detail.
The concentration C(x, t) of a pollutant in a semi-infinite pipe diffuses according to the PDE over 0<<o for > 0, where the diffusivity D is a given positive constant. An experiment is performed in which the initial concentration is zero everywhere, so that C(x, 0)0 for all r 20. Once t > 0 a fixed concentration of C(0, t) Co is imposed at x-0, where Co is a constant, and it is also assumed...
3.7 Underdamped free vibrations (e.g. a vibrating beam or slinky in damp air A system whose response is governed by the following constant-coefficient, is performing underdamped free vibrations if oc 1 (underdamped ). (Seetion 6.7). linear, 2td order, ODE ) and 1t)- (freei (a) After assuming a solution t)et where C and p are conustants, show every step that proves yit) can be expressed in terms of the yet-to-be-determined constants A and B and he constant wa (defined above) called...
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