Three traveling sinusoidal waves are on identical strings, with the same tension. The mathematical forms of the waves are y1(x, t) = ym sin(3x – 6t), y2(x, t) = ym sin(4x – 8t), and y3(x, t) = ym sin(6x – 12t), where x is in meters and t is in seconds. Match each mathematical form to the appropriate graph below.

Engineering

Physics

Standing Wave on String

Question

Three traveling sinusoidal waves are on identical strings, with the same tension. The mathematical forms of the waves are y₁(x, t) = y_m sin(3x – 6t), y₂(x, t) = y_m sin(4x – 8t), and y₃(x, t) = y_m sin(6x – 12t), where x is in meters and t is in seconds. Match each mathematical form to the appropriate graph below.

y₁ : iii, y₂ : ii, y₃ : i

y₁ : i, y₂ : ii, y₃ : iii

y₁ : ii, y₂ : i, y₃ : iii

y₁ : i, y₂ : iii, y₃ : ii

LiveFree JEE Main Previous Year Online Papers Live Free JEE Advance Previous Year Online Papers

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

The wave speed on a string is given by $v = \sqrt{\frac{T}{μ}}$ . Since tension T and linear density μ are identical for all waves, the speed v is constant. The standard wave form is $y = y_{m} \sin (k x - ω t)$ , where wave number k=2π/λ and angular frequency ω=2πf. The speed is also v=ω/k.

For each wave, calculate v = ω/k: $v_{1} = \frac{6}{3} = 2 v_{2} = \frac{8}{4} = 2 v_{3} = \frac{12}{6} = 2$ m/s. All have the same speed.

The graphs differ in wavelength. The wave number k is proportional to 1/λ. A larger k means a smaller wavelength. Comparing k values: y₁ has k=3, y₂ has k=4, y₃ has k=6. Therefore, y₃ has the smallest wavelength (graph i), y₂ has medium (graph ii), and y₁ has the largest (graph iii).

Final Answer: y₁ : iii, y₂ : ii, y₃ : i