The impedance for electromagnetic waves is defined as the ratio between the voltage and current waves,

where are inductance and capacitance per unit length in medium. For example, the inductance, capacitance, and impedance of a coaxial cable is given by whereThe fifth animation shows reflection and transmission of sinusoidal waves when the impedance of termianting load is one quarter of the cable impedance.

Reflection can be avoided entirely if a third medium having an impedance and length of quarter wavelength is inserted. This well known quarter wavelength impedance matching is shown in the last animation. Note that there are no standing waves and all wave energy is smoothly transferred to the cable 2.

V_g = 10 V, R_G = 50 Ohms, Z = 50 Ohms, R_L = 25 Ohms

Evolution of the voltage wave.

**with(plots):**

**animate(5*Heaviside(t-x)-5/3*Heaviside(t+x),x=-10..1,t=-9..11,color=red,numpoints=300,frames=100,view=[-10..0,0..5]);**

Evolution of the current wave.

**animate(.1*Heaviside(t-x)+.1/3*Heaviside(t+x),x=-10..1,t=-9..11,color=red,numpoints=300,frames=100,view=[-10..0,0..0.2]);**

V_g = 10 V, R_G = 0, Z = 50 Ohms, R_L = 25 Ohms

Voltage wave.

**animate(10*Heaviside(t-x)-10/3*Heaviside(t+x)+10/3*Heaviside(t-x-10)-10/9*Heaviside(t+x-10)+10/9*Heaviside(t-x-20)-10/27*Heaviside(t+x-20)+10/27*Heaviside(t-x-30),x=-5..0,t=-6..30,color=red,numpoints=300,frames=200,view=[-5..0,0..10]);**

Current wave.

` `**animate(.2*Heaviside(t-x)+.2/3*Heaviside(t+x)+.2/3*Heaviside(t-x-10)+.2/9*Heaviside(t+x-10)+.2/9*Heaviside(t-x-20)+.2/27*Heaviside(t+x-20)+.2/27*Heaviside(t-x-30),x=-5..0,t=-6..30,color=red,numpoints=300,frames=200,view=[-5..0,0..0.4]);**

Reflection of sinusoidal voltage wave when V_g = 1 V, Z_2/Z_1 = 1/4.
Note formation of incomplete standing in medium 1.

**with(plots):**

**animate((sin(.1*t-x)-3/5*sin(x+.1*t))*Heaviside(-x)+2/5*sin(.1*t-4*x)*Heaviside(x),x=-10..5,t=0..62,frames=30,color=red,numpoints=300);**

Impedance matching. Z_1 = 1, Z_2 = 1/4, Z_3 = 1/2. Inserted Z_3
medium
(shown in red) is quarter wavelength thick (or long). No reflection
occurs
and wave energy is smoothly transfered to cable 2. Note the absence of
standing waves.

**with(plots):**

**a:=animate(.75*(cos(2*Pi*.1*t-2*Pi*x)-0.25*cos(2*Pi*.1*t+2*Pi*x)),x=-0.25..0,t=0..20,color=red,frames=60):**

**b:=animate(cos(2*Pi*.1*t-2*Pi*x/2+Pi/4),x=-4..-0.25,t=0..20,color=blue,frames=60):**

**c:=animate(.5*cos(2*Pi*.1*t-4*Pi*x),x=0..2,t=0..20,color=green,frames=60):**

**display({a,b,c});**