EQUATION OF A STANDING WAVE
Plucking a string of a guitar you provoke a wave, whose equation is (neglecting the effect of friction and assuming that it is progressive):
y1=Asin[2π(t/T-x/λ)+α1]
This first wave propagates along the string until one of two fixed ends, where is reflected generating a second wave that propagates in the opposite direction with the same period, amplitude and wavelength. Since the wave is regressive, its equation is:
y2=Asin[2π(t/T+x/λ)+α2].
Then the motion of a point of the string is the result of the superposition of two waves: a progressive and a regressive one, for which the “true” distance from the equilibrium position is neither y1 or rio non è né y1 né, but
y=y1+y2=Asin[2π(t/T-x/λ)+α1]+Asin[2π(t/T+x/λ)+α2]
(the effects of the two waves are added algebraically).
In other words,
y=f(x,t)=A{sin[2π(t/T-x/λ)+α1]+sin[2π(t/T+x/λ)+α2]} (1),
where writing y=f(x,t) means that the value that of y depends on the x and time.
Since the string is fixed to its ends, they can not vibrate: then at least for x=0 must be y=0, not only in some particular instant (this happens in all the points of the string), but always. Ie, for any value of t (indicating with f(0,t) the value of y in the extreme where x=0, expressed as a function of the time t):
f(0,t)=A{sin[2πt/T+α1]+sin[2πt/T+α2]}.
This requires that the sines of the angles 2πt/T+α1 and 2πt/T+α2 are opposites, of course for each value of t. And that is exactly what happens to two angles that differ by a straight angle (180° or π radians). By choosing α1=0 for simplicity, it is then α2=π.
Therefore equation (1) becomes
y=f(x,t)=A{sin[2π(t/T-x/λ)]+sin[2π(t/T+x/λ)+π]},
and, imposing
p=2π(t/T+x/λ)+π=2πt/T+2πx/λ+π e
q=2π(t/T-x/λ)=2πt/T-2πx/λ,
f(x,t)=A{sinp+sinq}.
Recalling the first sum-to-product formula you obtain
f(x,t)=A{2sin[(p+q)/2]cos[(p-q)/2]}=2Asin[(p+q)/2]cos[(p-q)/2]. (2)
Now calculate the arguments of sine and cosine:
p+q=2πt/T+2πx/λ+2πt/T-2πx/λ=4πt/T+π;
(p+q)/2=2πt/T+π/2=(2π/T)·t+π/2.
p-q=2πt/T+2πx/λ-2πt/T+2πx/λ=4πx/λ+π;
(p-q)/2=2πx/λ+π/2.
Substituting these values in (2), and recalling the formula ω=2π/T:
f(x,t)=2Asin[(2π/T)·t+π/2]cos[2πx/λ+π/2]=2Acos[2πx/λ+π/2]sin[ωt+π/2]. (3)
The interesting thing is that the argument of sine depends on the time, but not on the position; vice versa that of cosine depends on x, but not on t. In any case the factor 2Acos[2πx/λ+π/2] depends only on x. If this amount is positive, you can put
R=2Acos[2πx/λ+π/2],
and (3) becomes
f(x,t)=Rsin[ωt+π/2].
But this is the equation of a simple harmonic motion: since R depends on x, while α0 is π/2 regardless of location, all the points for which cos[2πx/λ+π/2]>0 vibrate with simple harmonic motion, all in phase with each other and with variable amplitude depending on the position.
If instead cos[2πx/λ+π/2]<0, since both A (the amplitude of the wave) and R (the amplitude of the harmonic motion) can not be negative, put R=-2Acos[2πx/λ+π/2] (the opposite of a negative number is positive).
Exploit the opportunity to change twice the sign without changing the result:
f(x,t)=-2Acos[2πx/λ+π/2]{-sin[ωt+π/2]}=R{-sin[ωt+π/2]}.
As you have seen, the sines of angles that differ by a straight angle are opposed: you can then remove the minus sign in front of the sine, provided you subtract (or add, it would be the same!) π To the argument. Then
f(x,t)=Rsin[ωt+π/2-π]=Rsin[ωt-π/2].
Also in this case
all points vibrate with simple harmonic motion, in phase with each other and with variable amplitude depending on the position; however, there is a phase shift of a straight angle (it is said that they are in opposed phase)
from the previous.
Ultimately
all points vibrate in phase or in opposed phase between them, there is no longer a phase shift that varies gradually as the position. The wave is said a standing waves precisely as not propagates more.
Finally let’s examine some special cases: for simplicity, you refer to equation (3). The amplitude of the harmonic motion, since A is independent of x, depends only on the value of the cosine. As cosine is between -1 and 1, the minimum width is 0, and is obtained when cos[2πx/λ+π/2]=0. In this case 2πx/λ+π/2=π/2+kπ, where k is any integer (0, +1, -1, +2, -2, ...); then 2πx/λ=kπ, 2x=kλ and
x=k·(λ/2).
You have a node (a point that does not vibrate) if its distance from the origin of the string is a multiple of half a wavelength.
The amplitude of the harmonic motion is instead a maximum when the absolute value of the cosine has the maximum value, ie when cos[2πx/λ+π/2]=±1.
In this case 2πx/λ+π/2=kπ; then 2x/λ+1/2=k, 2x+λ/2=kλ, 2x=kλ-λ/2, x=k·(λ/2)-λ/4 and x=(2k-1)·(λ/4).
You have an antinode (a point that vibrates with maximum amplitude) when its distance from the origin of the string is an odd multiple of a quarter of a wavelength.