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tokens: be determined by scaling the weighted frequency. The quantities c S (), c A (), and c T () are the sound speed, Alfvén speed, and tube speed, respectively, and γ , k B , T 0 , μ , m H , B 0 , μ B , and ρ 0 are the specific heat ratio, Boltzmann’s constant, temperature, mean molecular weight, hydrogen atomic mass, magnetic field strength, magnetic permeability, and density, respectively. The subscript 0 indicates the equilibrium quantity. The () is the buoyancy frequency, where g = 0.274 km s −2 . The parameters Λ p () and Λ ρ () are the pressure and density scale heights, respectively. The prime indicates the derivative with respect to the vertical direction. As we assume the uniform vertical field in a non-isothermal stratified medium, and , giving . Hence the expression for the angular cutoff frequency is reduced towhere . Equation ( 2 ) describes umbral oscillations in the photosphere, accounting for the finite β effects. Equation ( 2 ) shows that f cutoff is directly related to Λ p and Λ ρ , which all can be determined from observations. In other words, and Λ ρ = γ Λ p . Thus, Equation ( 2 ) becomes a function of X . The sound speed c S is estimated from the plasma temperature. The temperature is obtained from the ratio of HMI continuum intensity to the quiet Sun’s, by applying the Planck function (Mathew et al. 2004 ). In other words, we solve the equation , where I , h , c , λ , k B , , and are the intensity ratio, Planck’s constant, speed of light, wavelength of Fe i line continuum (6172.1 Å), Boltzmann’s constant, the temperature at a given position, and the temperature of the quiet Sun (6000 K), respectively. Using μ = 1.3 and γ = 5/3, we obtain the sound speed as km s −1 . We look for a positive solution for X of Equation ( 2 ) by changing the scaling of in the range below 1.0. We found that positive solutions for all pixels inside the umbra exist when the scaling is in 0.5–0.55. Thus we take 0.55 as the upper limit of the scaling. The solution allows us to calculate the Alfvén speed, using the expressions ), the plasma- β , as , and the mass density as for each pixel of the umbra. The magnetic field strength B 0 is obtained as the de-projected HMI magnetogram, which may correspond to the magnetic field slightly above the height of continuum formation (Norton et al. 2006 ). Slightly different values of equilibrium quantities do not give us significantly different results. For example, when δT 0 = ±100 K, and are less than 50 m s −1 , and is less than 0.02 for the given sunspot. When we choose δλ = −2 Å, the absolute values of deviation of speeds and plasma- β are less than 0.1 m s −1 and 10 −4 , respectively. 3. Results We illustrate the application of the developed method to a sunspot located near the disk center, observed during 00:00–01:00 UT on 2014 August 28, in NOAA 12149. The center of a sunspot is tracked synodically, accounting for the effect of the solar differential rotation. We used the differential rotation law, in μ rad