Stationary Waves

(1) Formation:: $y_1 = a sin(wt – {2πx}/{λ})$ & $y_2 = a sin(wt + {2πx}/{λ})$
$y = A sinwt$ … $A = 2a cos {2πx}/{λ}$

(2) $y = 2a cos {2πx}/{λ} sin {2πt}/{T} $

(3) Distance between nodes/antinodes = λ/2
Distance between nodes & antinodes = λ/4

(4) $V = √{{T}/{m}}$ m -> mass per unit length

(5) (String)
$l = λ/2$ ; $n_o = 1/{2l} √{{T}/{m}} $

(6) Overtones: $n_p = {P+1}/{2}$

(7) Vibration of one end closed air column: l = λ/4
$n_o = 1/{4l} √{{T}/{m}}$
Overtones: $n_p = {2P + 1}/{l}$

(8) Resonance Tube:
$n_o = {v}/{4 l_1}$ & $n_1 = {3v}/{4 l_2}$

$e = {n_2 l_2 – n_1 l_1}/{n_1 – n_2} \text" & " n_2 = n_1/3$

$e = {l_2 – 3 l_1}/{2}$ ; $λ = 2(l_2 – l_1)$ & $v = 2n (l_2 – l_1)$

(9) Melde’s experiment: l = L/P … length of one loop

Parallel position: $n = P/L √{{T}/{m}}$ (longitudinal)

Perpendicular position: $n = P/{2L} √{{T}/{m}}$ (transverse)

(10) $n α √{273° + t}$

(11) $TP^2 = constant$

(12) $m = πr^2 ρ$ … m -> linear density

(13) e = 0.3 d

(14) ${ρ_b}/{ρ_l – ρ_b}$