An Allusion to Rupert Brooke in Tolkien?

In addressing the assumptions of some readers that The Lord of the Rings was about World War II, Tolkien reminds us that he had been in World War I (FR xxiv, emphasis added):

One has indeed personally to come under the shadow of war to feel fully its oppression; but as the years go by it seems now often forgotten that to be caught in youth by 1914 was no less hideous an experience than to be involved in 1939 and the following years. By 1918 all but one of my close friends were dead. 

In John Garth's Mythgard class on Tolkien and the Great War today we were reading Rupert Brooke's 1914 sonnet Peace and a phrase leaped out at me (emphasis added):

Now, God be thanked Who has matched us with His hour,
And caught our youth, and wakened us from sleeping,
With hand made sure, clear eye, and sharpened power,
To turn, as swimmers into cleanness leaping,
Glad from a world grown old and cold and weary,
Leave the sick hearts that honour could not move,
And half-men, and their dirty songs and dreary,
And all the little emptiness of love! 

Oh! we, who have known shame, we have found release there,
Where there's no ill, no grief, but sleep has mending,
Naught broken save this body, lost but breath;
Nothing to shake the laughing heart's long peace there
But only agony, and that has ending;
And the worst friend and enemy is but Death.

Even if the striking similarity of phrase is accidental -- which I don't believe for a moment, given Brooke's fame early and late as the poster boy of World War I poets -- Tolkien and Brooke take the idea of youth being caught in such different directions that I think I will have to give these passages further thought.  But not tonight.  Tonight I just want to revel in the pleasure of having heard the echo.