Two superimposing waves are represented by the equations y₁ = 2sin(2π(10t − 0.4x)) and y₂ = 4sin(2π(20t − 0.8x)). The ratio of lₘₐₓ to lᵢₘ is:

• A: 36:4
• B: 25:9
• C: 1:4
• D: 4:1

To find the ratio of the maximum amplitude (lₘₐₓ) to the minimum amplitude (lᵢₘ) of the superimposing waves given by the equations y₁ and y₂, we first identify the amplitudes of each wave.

Identifying Amplitudes

The amplitude of the first wave, y₁ = 2sin(2π(10t − 0.4x)), is 2. The amplitude of the second wave, y₂ = 4sin(2π(20t − 0.8x)), is 4.

Calculating Maximum and Minimum Amplitudes

• Maximum Amplitude (lₘₐₓ): This is the sum of the amplitudes of both waves when they are in phase. Thus, lₘₐₓ = 2 + 4 = 6.
• Minimum Amplitude (lᵢₘ): This occurs when the waves are out of phase. Therefore, lᵢₘ = |2 - 4| = 2.

Finding the Ratio

Now, we can find the ratio of lₘₐₓ to lᵢₘ:

Ratio = lₘₐₓ : lᵢₘ = 6 : 2 = 3 : 1.

Comparing with Options

None of the provided options (A: 36:4, B: 25:9, C: 1:4, D: 4:1) match the calculated ratio of 3:1. Therefore, it seems there may be an error in the options given.