傅里叶变换与钟形频率示例：求解b部分遇到瓶颈

Hey there! Let's break through this stuck part together — you've already laid a great foundation, so we just need to connect the dots.

First, let's recap what you've got right so far:

• You correctly calculated I(ω₀) = 1/(2πK²), so 1% of that intensity is 1/(200πK²).
• Your derivation of (ω - ω₀)²/ω₀² = (0.5/100)² (simplifying to |ω - ω₀| = 0.005ω₀) aligns perfectly with the problem's condition about frequency deviation and intensity drop.

Next Steps to Solve Part (b) #

Assuming the intensity vs. frequency distribution follows a Lorentzian curve (standard for resonant systems, which this problem likely involves), the general form is:

I(ω) = I(ω₀) / [1 + ((ω - ω₀)² / σ²)]

where σ is a characteristic width parameter tied to your constant K.

1. Plug in the critical condition
   We know when |ω - ω₀| = 0.005ω₀, I(ω) = 0.01I(ω₀). Substitute these into the Lorentzian equation:

   0.01I(ω₀) = I(ω₀) / [1 + ((0.005ω₀)² / σ²)]
   

2. Simplify to solve for σ
   Cancel out I(ω₀) from both sides, then take the reciprocal of each term:

   100 = 1 + (0.000025ω₀² / σ²)
   

   Subtract 1 from both sides:

   99 = 0.000025ω₀² / σ²
   

   Rearrange to solve for σ²:

   σ² = (0.000025ω₀²) / 99
   

   Take the square root:

   σ = (0.005ω₀) / √99 ≈ 0.0005025ω₀
   

3. Connect σ to your constant K
   From part (a), your expression for I(ω₀) implies σ is related to K (typically from resonant response derivations). If we use the standard relation σ = K/(2ω₀) (adjust if your part (a) derivation uses a different scaling), substitute σ into this:

   0.0005025ω₀ = K/(2ω₀)
   

   Solve for K:

   K ≈ 0.001005ω₀² ≈ ω₀²/995
   

If your part (a) used a different form of the intensity equation (e.g., with a (2Q(ω-ω₀)/ω₀)² term instead), the core logic stays the same: substitute the critical frequency and intensity values into the equation, then solve for the unknown parameter (like K or quality factor Q).

内容的提问来源于stack exchange，提问作者DJA