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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "PHYSICS RESOURCE PACKETS P
ROJECT                               File: DOPPL.MWS" }}{PARA 0 "" 0 "
" {TEXT -1 110 "Rose-Hulman Institute of Technology                   \+
                    Authors: Perry Peters & Greg Williby" }}{PARA 0 "
" 0 "" {TEXT -1 97 "http://www.rose-hulman.edu/~moloney               \+
                    Software: Maple V Release 4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 226 "Assume a source radiatin
g radially and draw 10 crests (circles) of the sound waves.  Show the \+
position of the wave crests for an object moving near the speed of sou
nd, at the speed of sound, and faster than the speed of sound." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest
art;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Load the plots package." 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 74 "Define the function \"draw\" so that it will retur
n A for A>0 and 0 for A<0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "draw:
=A->A*Heaviside(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1169 "The loop
 below will generate 11 parametric equations (h0,h1,h2,...,h10).  h0 d
escribes the motion of the first sound wave emitted, h1 describes the \+
motion of the second wave and so on.  The parametric equations are of \+
the form [x(n),y(n),n=a..b].  We want to generate circles that radiate
 outward from the sound source at the speed of sound, so there is a co
sine term in x(n), and a sine term in y(n).  343*j*(0.75) is the locat
ion of the center of the circle being drawn.  In the animate command a
ll 11 parametric equations are ploted on the same graph at the same ti
me for t=0 to 11 seconds.  The object moves along at a speed of 343*0.
75 m/s and (for clarity) emits only one wavefront every second.  The \+
\"draw\" function controls the number and size of the circles drawn.  \+
When 0>t>1, the only time (t-j) is positive in the draw function is wh
en j=0, so only the contents of h0 are visible in the animation.  When
 1>t>2, h0 is still drawn, but now the circle in h1 is also visible.  \+
The radius of the first circle is 343*(t-0), and the radius of the sec
ond circle is 343*(t-1).  More of the circles will become visible as t
ime progresses until they all are when t>10." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "for j from 0 to 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 84 "h.j:=[343*draw(t-j)*cos(theta)+343*j*(0.75),343*draw(t-j)*sin(
theta),theta=0..2*Pi];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Animate h0..h10 for t=0..11 s." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "animate(\{h.(0..10)\},t=0..11,title
=`Mach 0.75`,color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Now i
ncrease the speed of the source from 343*0.75 m/s to 343 m/s." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 0 to 10 do" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 81 "h.j:=[343*draw(t-j)*cos(theta)+343*j*(1),343*
draw(t-j)*sin(theta),theta=0..2*Pi];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animate(\{h.(0.
.10)\},t=0..11,title=`Mach 1`,color=red);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 32 "Increase the speed to 2*343 m/s." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "for j from 0 to 10 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 81 "h.j:=[343*draw(t-j)*cos(theta)+343*j*(2),343*draw(t-j)*sin(the
ta),theta=0..2*Pi];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animate(\{h.(0..10)\},t=0..1
1,title=`Mach 2`,color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "A
ccelerate the source from Mach 0.75 to Mach 1.25 in 10 seconds." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 0 to 10 do" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 89 "h.j:=[343*draw(t-j)*cos(theta)+343*j*(0.75+j/
20),343*draw(t-j)*sin(theta),theta=0..2*Pi];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "anim
ate(\{h.(0..10)\},t=0..11,title=`Accelerating from Mach 0.75 to Mach 1
.25`,color=red);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Instead of an
imating the wavefronts as a function of time,animate them as a functio
n of speed." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 1 to 11 d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "h.j:=[343*j*cos(theta)-343*j*(
speed),343*j*sin(theta),theta=0..2*Pi];" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "animate(\{h.(
1..11)\},speed=0..2,title=`Shape of wavefronts from Mach 0 to Mach 2`,
color=red);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "
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