Under Section 1.4 of the lecture notes, Equation 1.8 defines a wave packet as the infinite sum of waves consisting of infinitesimally differing wave numbers. In practice, however, the difference in the wave number is finite and so is the sum of the waves. Suppose a wave packet is created by adding N+ 1 sinc waves with differing angular frequency dw. That is, (N+1) Swt sin V (t) 2sin (wo + now)t] Sw (1) sin Wo + dwt sin n=0

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Under Section 1.4 of the lecture notes, Equation 1.8 defines a wave packet as the infinite sum of
waves consisting of infinitesimally differing wave numbers. In practice, however, the difference
in the wave number is finite and so is the sum of the waves. Suppose a wave packet is created
by adding N+1 sine waves with differing angular frequency dw. That is,
sin
(N+1)
Swt
V(t) = sin [(wo + ndw)t]
(1)
sin
Wo t
Swt
sin
n=0
Transcribed Image Text:Under Section 1.4 of the lecture notes, Equation 1.8 defines a wave packet as the infinite sum of waves consisting of infinitesimally differing wave numbers. In practice, however, the difference in the wave number is finite and so is the sum of the waves. Suppose a wave packet is created by adding N+1 sine waves with differing angular frequency dw. That is, sin (N+1) Swt V(t) = sin [(wo + ndw)t] (1) sin Wo t Swt sin n=0
7. Prove equation (1). Hint: Use one of the trigonometric identities for sin and use
(N+1)0
(N+1)0
sin
NO
sin
NO
and cos(n0)
COS
sin(no)
sin
sin ()
sin ()
T=0
n=0
Transcribed Image Text:7. Prove equation (1). Hint: Use one of the trigonometric identities for sin and use (N+1)0 (N+1)0 sin NO sin NO and cos(n0) COS sin(no) sin sin () sin () T=0 n=0
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