Solving dispersion equation

You can use an apply function, like so:

# Parameters
g <- 9.81
h <- 8
w <- c(0.1, 0.2, 0.3)

# Function to find root of
disp_root <- function(k, w) {k * g * tanh(k * h) - w^2 }

# Apply for each w
res <- sapply(w, function(x)rootSolve::uniroot.all(disp_root, c(-1,1), w = x))

# Repackage results
df_res <- data.frame(w, t(res))

# Fix names
names(df_res)[2:3] <- c("first_root", "second_root")

# Examine results
df_res
#>     w  first_root second_root
#> 1 0.1 -0.01126658  0.01126658
#> 2 0.2 -0.02272161  0.02272161
#> 3 0.3 -0.03419646  0.03419646

^{Created on 2020-01-31 by the reprex package (v0.3.0)}