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Cards (32)
What is the title of the presentation?
EG-232
: Multivariable
Calculus
for Medical Engineers
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What is the main focus of the PDEs section?
Constructing and solving PDEs for
physical processes
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What are the two types of PDEs mentioned?
Wave equation
and
heat equation
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What does the wave equation represent in 3D?
Sound waves and displacement of air molecules
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What does the 1D wave equation represent?
Waves on a liquid surface or displacement of a wire
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What law is applied to form the wave equation?
Newton's 2nd law
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What does Hooke's law relate to in the wave equation?
The force on mass from adjacent springs
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What happens to a discrete system of masses/springs for small h?
It
becomes
a
continuous system
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What is the formula for density in the wave equation?
Density, r = m/
Ah
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What does the spring constant k equal in the wave equation?
k =
EA
/h
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What does the expression inside the large brackets equal in the wave equation?
The
2nd derivative
of u with respect to
position
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What type of waves does the derived 1D wave equation represent?
Longitudinal
(
sound
) waves in a solid
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What is the form of displacement u(x,t) that indicates wave properties?
A
pulse
moving
from
left
to
right
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What does the constant c represent in the wave equation?
The
wave speed
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What assumption is made about the wave in the wave equation?
There are no
losses
; it carries on forever
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What are the boundary conditions for a flexible string in the wave equation?
Fixed endpoints
at
x=0
and
x=L
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What are the initial conditions for the wave equation of a string?
Initial deflection
f(x)
and initial velocity
g(x)
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What does separation of variables allow in solving PDEs?
Reducing a PDE to easier
ODEs
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What form do the solutions take when using separation of variables?
Factorized solutions
X(x)
and
T(t)
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What happens when separating variables in the wave equation?
It must equal a
constant
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What type of functions are solutions to the wave equation?
Combinations of
sines
and
cosines
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What does k represent in the context of the wave equation?
It
can
be
positive
,
negative
, or
zero
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What is the general solution for second-order homogeneous ODEs?
u =
A1e
^(
m1x
) +
A2e
^(
m2x
)
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What is the auxiliary equation used for in second-order linear ODEs?
To find roots for the
general solution
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What does the Helmholtz equation relate to in the wave equation?
It is a
constant
not a function of
x
or t
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What are eigenvalues and eigenfunctions in the wave equation?
Discrete
solutions
with
corresponding
eigenvalues
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How does the wave equation relate to Fourier Series?
It
reminds
us
of
Fourier Series
solutions
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What are the steps involved in solving the wave equation using separation of variables?
Assume
factorized solutions
X(x) and T(t)
Rewrite the wave equation
Separate variables to equal a constant
Solve the resulting
ordinary differential equations
Combine solutions to form the general solution
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What are the boundary and initial conditions for a vibrating string?
Boundary conditions: Fixed endpoints at
x=0
and
x=L
Initial conditions:
Initial
deflection
f(x) at
t=0
Initial
velocity
g(x) at t=0
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What is the significance of the wave speed constant c in the wave equation?
Represents the speed of wave
propagation
Relates to waves traveling in both directions
Assumes no energy loss during propagation
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What are the implications of k being positive, negative, or zero in the wave equation?
Positive k
: Oscillatory solutions
Negative k
:
Non-oscillatory
solutions
Zero k
: Static solutions
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What is the relationship between the wave equation and the concept of eigenvalues and eigenfunctions?
Eigenvalues correspond to discrete solutions
Each eigenvalue has a unique eigenfunction
Solutions are determined by
boundary conditions
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